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Class 10 Maths Chapter 5 – Arithmetic Progression

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Exercise 5.1 (Page 99)

Question 1:
In which of the following situations, does the list of numbers involved make an arithmetic progression and why?
(i) The taxi fare after each km when the fare is ₹ 15 for the first km and ₹ 8 for each additional km.

(ii) The amount of air present in a cylinder when a vacuum pump removes 14 of the air remaining in the cylinder at a time.
(iii) The cost of digging a well after every meter of digging, when it costs ₹ 150 for the first meter and rises by ₹ 50 for each subsequent meter.

(iv) The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8% per annum.

Answer 1:
(i) We can write the given condition as;
Taxi fare for 1 km = 15
Taxi fare for first 2 kms = 15+8 = 23
Taxi fare for first 3 kms = 23+8 = 31
Taxi fare for first 4 kms = 31+8 = 39
And so on……

Thus, 15, 23, 31, 39 … forms an A.P. because every next term is 8 more than the preceding term.

(ii) Let the volume of air in a cylinder, initially, be V litres.

In each stroke, the vacuum pump removes 1/4th of air remaining in the cylinder at a time. Or we can say, after every
stroke, 1-1/4 = 3/4th part of air will remain.

Therefore, volumes will be V, 3V/4 , (3V/4)2 , (3V/4)3…and so on
Clearly, we can see here, the adjacent terms of this series do not have the common difference between them. Therefore, this series is not an A.P.

(iii) We can write the given condition as;

Cost of digging a well for first metre = Rs.150
Cost of digging a well for first 2 metres = Rs.150+50 = Rs.200
Cost of digging a well for first 3 metres = Rs.200+50 = Rs.250
Cost of digging a well for first 4 metres =Rs.250+50 = Rs.300

And so on..
Clearly, 150, 200, 250, 300 … forms an A.P. with a common difference of 50 between each term.

(iv) We know that if Rs. P is deposited at r% compound interest per annum for n years, the amount of money will be:

P(1+r/100)n
Therefore, after each year, the amount of money will be;

10000(1+8/100), 10000(1+8/100)2, 10000(1+8/100)3……
Clearly, the terms of this series do not have the common difference between them. Therefore, this is not an A.P.

Question 2:
Write first four terms of the A.P. when the first term a and the common difference are given as follows:
(i) a = 10, d = 10
(ii) a = -2, d = 0
(iii) a = 4, d = – 3
(iv) a = -1 d = 1/2
(v) a = – 1.25, d = – 0.25

Answer 2:
(i) a = 10, d = 10
Let us consider, the Arithmetic Progression series be a1, a2, a3, a4, a5 …

a1 = a = 10
a2 = a1+d = 10+10 = 20
a3 = a2+d = 20+10 = 30
a4 = a3+d = 30+10 = 40
a5 = a4+d = 40+10 = 50
And so on…

Therefore, the A.P. series will be 10, 20, 30, 40, 50 …
And First four terms of this A.P. will be 10, 20, 30, and 40.

(ii) a = – 2, d = 0
Let us consider, the Arithmetic Progression series be a1, a2, a3, a4, a5 …

a1 = a = -2
a2 = a1+d = – 2+0 = – 2
a3 = a2+d = – 2+0 = – 2
a4 = a3+d = – 2+0 = – 2

Therefore, the A.P. series will be – 2, – 2, – 2, – 2 …
And, First four terms of this A.P. will be – 2, – 2, – 2 and – 2.

(iii) a = 4, d = – 3
Let us consider, the Arithmetic Progression series be a1, a2, a3, a4, a5 …

a1 = a = 4
a2 = a1+d = 4-3 = 1
a3 = a2+d = 1-3 = – 2
a4 = a3+d = -2-3 = – 5

Therefore, the A.P. series will be 4, 1, – 2 – 5 …
And, first four terms of this A.P. will be 4, 1, – 2 and – 5.

(iv) a = – 1, d = 1/2
Let us consider, the Arithmetic Progression series be a1, a2, a3, a4, a5 …

a2 = a1+d = -1+1/2 = -1/2
a3 = a2+d = -1/2+1/2 = 0
a4 = a3+d = 0+1/2 = 1/2

Thus, the A.P. series will be-1, -1/2, 0, 1/2
And First four terms of this A.P. will be -1, -1/2, 0 and 1/2.

(v) a = – 1.25, d = – 0.25
Let us consider, the Arithmetic Progression series be a1, a2, a3, a4, a5 …

a1 = a = – 1.25
a2 = a1 + d = – 1.25-0.25 = – 1.50
a3 = a2 + d = – 1.50-0.25 = – 1.75
a4 = a3 + d = – 1.75-0.25 = – 2.00

Therefore, the A.P series will be 1.25, – 1.50, – 1.75, – 2.00 ……..
And first four terms of this A.P. will be – 1.25, – 1.50, – 1.75 and – 2.00.

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Question 3:
For the following A.P.s, write the first term and the common difference.
(i) 3, 1, – 1, – 3 …
(ii) -5, – 1, 3, 7 …
(iii) 1/3, 5/3, 9/3, 13/3 ….
(iv) 0.6, 1.7, 2.8, 3.9 …

Answer 3:
(i) Given series,
3, 1, – 1, – 3 …

First term, a = 3
Common difference, d = Second term – First term

⇒ 1 – 3 = -2
⇒ d = -2

(ii) Given series, – 5, – 1, 3, 7 …
First term, a = -5

Common difference, d = Second term – First term
⇒ ( – 1)-( – 5) = – 1+5 = 4

(iii) Given series, 1/3, 5/3, 9/3, 13/3 ….
First term, a = 1/3
Common difference, d = Second term – First term
⇒ 5/3 – 1/3 = 4/3

(iv) Given series, 0.6, 1.7, 2.8, 3.9 …
First term, a = 0.6
Common difference, d = Second term – First term

⇒ 1.7 – 0.6
⇒ 1.1

Question 4:
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.

(i) 2, 4, 8, 16 …
(ii) 2, 5/2, 3, 7/2 ….
(iii) -1.2, -3.2, -5.2, -7.2 …
(iv) -10, – 6, – 2, 2 …
(v) 3, 3 + √2, 3 + 2√2, 3 + 3√2
(vi) 0.2, 0.22, 0.222, 0.2222 ….
(vii) 0, – 4, – 8, – 12 …
(viii) -1/2, -1/2, -1/2, -1/2 ….
(ix) 1, 3, 9, 27 …
(x) a, 2a, 3a, 4a …
(xi) a, a², a³, a⁴ …
(xii) √2, √8, √18, √32 …
(xiii) √3, √6, √9, √12 …
(xiv) 1², 3², 5², 7² …
(xv) 1², 5², 7², 7³ …

Answer 4:
(i) Given to us,
2, 4, 8, 16 …

Here, the common difference is;

a₂ – a₁ = 4 – 2 = 2
a₃ – a₂ = 8 – 4 = 4
a₄ – a₃ = 16 – 8 = 8

Since, a_n+1 – a_n or the common difference is not the same every time.
Therefore, the given series are not forming an A.P.

ii) Given, 2, 5/2, 3, 7/2 ….
Here,

a₂ – a₁ = 5/2-2 = 1/2
a₃ – a₂ = 3-5/2 = 1/2
a₄ – a₃ = 7/2-3 = 1/2

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = 1/2 and the given series are in A.P.

The next three terms are;

a₅ = 7/2+1/2 = 4
a₆ = 4 +1/2 = 9/2
a₇ = 9/2 +1/2 = 5

(iii) Given, -1.2, – 3.2, -5.2, -7.2 …
Here,

a₂ – a₁ = (-3.2)-(-1.2) = -2
a₃ – a₂ = (-5.2)-(-3.2) = -2
a₄ – a₃ = (-7.2)-(-5.2) = -2

Since, a_n+1 – a_n or common difference is same every time.
Therefore, d = -2 and the given series are in A.P.

Hence, next three terms are;

a₅ = – 7.2-2 = -9.2
a₆ = – 9.2-2 = – 11.2
a₇ = – 11.2-2 = – 13.2

(iv) Given, -10, – 6, – 2, 2 …
Here, the terms and their difference are;

a₂ – a₁ = (-6)-(-10) = 4
a₃ – a₂ = (-2)-(-6) = 4
a₄ – a₃ = (2 -(-2) = 4

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = 4 and the given numbers are in A.P.

Hence, next three terms are;

a₅ = 2+4 = 6
a₆ = 6+4 = 10
a₇ = 10+4 = 14

(v) Given, 3, 3+√2, 3+2√2, 3+3√2
Here,

a₂ – a₁ = 3+√2-3 = √2
a₃ – a₂ = (3+2√2)-(3+√2) = √2
a₄ – a₃ = (3+3√2) – (3+2√2) = √2

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = √2 and the given series forms a A.P.

Hence, next three terms are;

a₅ = (3+√2) +√2 = 3+4√2
a₆ = (3+4√2)+√2 = 3+5√2
a₇ = (3+5√2)+√2 = 3+6√2

(vi) 0.2, 0.22, 0.222, 0.2222 ….
Here,

a₂ – a₁ = 0.22-0.2 = 0.02
a₃ – a₂ = 0.222-0.22 = 0.002
a₄ – a₃ = 0.2222-0.222 = 0.0002

Since, a_n+1 – a_n or the common difference is not same every time.
Therefore, and the given series doesn’t forms a A.P.

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(vii) 0, -4, -8, -12 …
Here,

a₂ – a₁ = (-4)-0 = -4
a₃ – a₂ = (-8)-(-4) = -4
a₄ – a₃ = (-12)-(-8) = -4

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = -4 and the given series forms a A.P.

Hence, next three terms are;

a₅ = -12-4 = -16
a₆ = -16-4 = -20
a₇ = -20-4 = -24

(viii) -1/2, -1/2, -1/2, -1/2 ….
Here,

a₂ – a₁ = (-1/2) – (-1/2) = 0
a₃ – a₂ = (-1/2) – (-1/2) = 0
a₄ – a₃ = (-1/2) – (-1/2) = 0

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = 0 and the given series forms a A.P.

Hence, next three terms are;

a₅ = (-1/2)-0 = -1/2
a₆ = (-1/2)-0 = -1/2
a₇ = (-1/2)-0 = -1/2

(ix) 1, 3, 9, 27 …
Here,

a₂ – a₁ = 3-1 = 2
a₃ – a₂ = 9-3 = 6
a₄ – a₃ = 27-9 = 18

Since, a_n+1 – a_n or the common difference is not same every time.
Therefore, and the given series doesn’t form a A.P.

(x) a, 2a, 3a, 4a …
Here,

a₂ – a₁ = 2a–a = a
a₃ – a₂ = 3a-2a = a
a₄ – a₃ = 4a-3a = a

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = a and the given series forms a A.P.

Hence, next three terms are;

a₅ = 4a+a = 5a
a₆ = 5a+a = 6a
a₇ = 6a+a = 7a

(xi) a, a², a³, a⁴ …
Here,

a₂ – a₁ = a²–a = a(a-1)
a₃ – a₂ = a³ –a² = a²(a-1)
a₄ – a₃ = a⁴ – a³ = a³(a-1)

Since, a_n+1 – a_n or the common difference is not same every time.
Therefore, the given series doesn’t forms a A.P.

(xii) √2, √8, √18, √32 …
Here,

a₂ – a₁ = √8-√2  = 2√2-√2 = √2
a₃ – a₂ = √18-√8 = 3√2-2√2 = √2
a₄ – a₃ = 4√2-3√2 = √2

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = √2 and the given series forms a A.P.

Hence, next three terms are;

a₅ = √32+√2 = 4√2+√2 = 5√2 = √50
a₆ = 5√2+√2 = 6√2 = √72
a₇ = 6√2+√2 = 7√2 = √98

(xiii) √3, √6, √9, √12 …
Here,

a₂ – a₁ = √6-√3 = √3×√2-√3 = √3(√2-1)
a₃ – a₂ = √9-√6 = 3-√6 = √3(√3-√2)
a₄ – a₃ = √12 – √9 = 2√3 – √3×√3 = √3(2-√3)

Since, a_n+1 – a_n or the common difference is not same every time.
Therefore, the given series doesn’t form a A.P.

(xiv) 1², 3², 5², 7² …
Or, 1, 9, 25, 49 …..
Here,

a₂ − a₁ = 9−1 = 8
a₃ − a₂ = 25−9 = 16
a₄ − a₃ = 49−25 = 24

Since, a_n+1 – a_n or the common difference is not same every time.
Therefore, the given series doesn’t form a A.P.

(xv) 1², 5², 7², 73 …
Or 1, 25, 49, 73 …
Here,

a₂ − a₁ = 25−1 = 24
a₃ − a₂ = 49−25 = 24
a₄ − a₃ = 73−49 = 24

Since, a_n+1 – a_n or the common difference is same every time.
Therefore, d = 24 and the given series forms a A.P.

Hence, next three terms are;

a₅ = 73+24 = 97
a₆ = 97+24 = 121
a₇ = 121+24 = 145

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Exercise 5.2 (Page 105)

Question 1:
Fill in the blanks in the following table, given that a is the first term, d the common difference and a_n the n^th term of the A.P.

Answer 1:
(i) a = 7, d = 3, n = 8, a_n = ?
We know that,
For an A.P. a_n = a + (n − 1) d
= 7 + (8 − 1) 3
= 7 + (7) 3
= 7 + 21 = 28
Hence, a_n = 28

(ii) Given that
a = −18, n = 10, a_n = 0, d = ?
We know that,
a_n = a + (n − 1) d
0 = − 18 + (10 − 1) d
18 = 9d
d = 18/9 = 2
Hence, common difference, d = 2

(iii) Given that
d = −3, n = 18, a_n = −5
We know that,
a_n = a + (n − 1) d
−5 = a + (18 − 1) (−3)
−5 = a + (17) (−3)
−5 = a − 51
a = 51 − 5 = 46
Hence, a = 46

(iv) a = −18.9, d = 2.5, a_n = 3.6, n = ?
We know that,
a_n = a + (n − 1) d
3.6 = − 18.9 + (n − 1) 2.5
3.6 + 18.9 = (n − 1) 2.5
22.5 = (n − 1) 2.5
(n – 1) = 22.5/2.5
n – 1 = 9
n = 10
Hence, n = 10

(v) a = 3.5, d = 0, n = 105, a_n = ?
We know that,
a_n = a + (n − 1) d
a_n = 3.5 + (105 − 1) 0
a_n = 3.5 + 104 × 0
a_n = 3.5
Hence, a_n = 3.5

Question 2:
Choose the correct choice in the following and justify:
(i) 30th term of the A.P: 10,7, 4, …, is
(A) 97 (B) 77 (C) −77 (D) −87

(ii) 11th term of the A.P. -3, -1/2, ,2 …. is
(A) 28 (B) 22 (C) – 38 (D)-481/2

Answer 2:
(i) Given that
A.P. 10, 7, 4, …
First term, a = 10
Common difference, d = a₂ − a₁ = 7 − 10 = −3
We know that, a_n = a + (n − 1) d
a₃₀ = 10 + (30 − 1) (−3)
a₃₀ = 10 + (29) (−3)
a₃₀ = 10 − 87 = −77
Hence, the correct answer is option C.

(ii) Given that A.P. is -3, -1/2, ,2 …
First term a = – 3
Common difference, d = a₂ − a₁ = (-1/2) – (-3)
= (-1/2) + 3 = 5/2
We know that,a_n = a + (n − 1) d
a₁₁ = 3 + (11 -1)(5/2)
a₁₁ = 3 + (10)(5/2)
a₁₁ = -3 + 25
a₁₁ = 22
Hence, the answer is option B.

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Question 3:
In the following APs find the missing term in the boxes.

Answer 3:
(i) For this A.P.,a = 2
a₃ = 26
We know that, a_n = a + (n − 1) d
a₃ = 2 + (3 – 1) d
26 = 2 + 2d
24 = 2d
d = 12
a₂ = 2 + (2 – 1) 12
= 14
Therefore, 14 is the missing term.

(ii) For this A.P.,
a₂ = 13 and
a₄ = 3
We know that, a_n = a + (n − 1) d
a₂ = a + (2 – 1) d
13 = a + d … (i)
a₄ = a + (4 – 1) d
3 = a + 3d … (ii)
On subtracting (i) from (ii), we get
– 10 = 2d
d = – 5

From equation (i), we get
13 = a + (-5)
a = 18
a₃ = 18 + (3 – 1) (-5)
= 18 + 2 (-5) = 18 – 10 = 8
Therefore, the missing terms are 18 and 8 respectively.

(iii) For this A.P.,
a = 5 and
a₄ = 19/2
We know that, a_n = a + (n − 1) d
a₄ = a + (4 – 1) d
19/2 = 5 + 3d
19/2 – 5 = 3d3d = 9/2
d = 3/2

a₂ = a + (2 – 1) d
a₂ = 5 + 3/2
a₂ = 13/2
a₃ = a + (3 – 1) d
a₃ = 5 + 2×3/2a₃ = 8Therefore, the missing terms are 13/2 and 8 respectively.

(iv) For this A.P.,
a = −4 and
a₆ = 6
We know that,
a_n = a + (n − 1) d
a₆ = a + (6 − 1) d
6 = − 4 + 5d
10 = 5d
d = 2
a₂ = a + d = − 4 + 2 = −2
a₃ = a + 2d = − 4 + 2 (2) = 0
a₄ = a + 3d = − 4 + 3 (2) = 2
a₅ = a + 4d = − 4 + 4 (2) = 4
Therefore, the missing terms are −2, 0, 2, and 4 respectively.

(v)
For this A.P.,
a₂ = 38
a₆ = −22
We know that
a_n = a + (n − 1) d
a₂ = a + (2 − 1) d
38 = a + d … (i)
a₆ = a + (6 − 1) d
−22 = a + 5d … (ii)
On subtracting equation (i) from (ii), we get
− 22 − 38 = 4d
−60 = 4d
d = −15
a = a₂ − d = 38 − (−15) = 53
a₃ = a + 2d = 53 + 2 (−15) = 23
a₄ = a + 3d = 53 + 3 (−15) = 8
a₅ = a + 4d = 53 + 4 (−15) = −7
Therefore, the missing terms are 53, 23, 8, and −7 respectively.

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Question 4:
Which term of the A.P. 3, 8, 13, 18, … is 78?

Answer 4:
Given the A.P. series as3, 8, 13, 18, …
First term, a = 3
Common difference, d = a₂ − a₁ = 8 − 3 = 5

Let the n^th term of given A.P. be 78. Now as we know,
a_n = a+(n−1)d

Therefore,

78 = 3+(n −1)5
75 = (n−1)5
(n−1) = 15
n = 16

Hence, 16^th term of this A.P. is 78.

Question 5:
Find the number of terms in each of the following A.P.

(i) 7, 13, 19, …, 205
(ii) 18,151/2,13,-47

Answer 5:

(i) For this A.P.,
a = 7
d = a₂ − a₁ = 13 − 7 = 6
Let there are n terms in this A.P.
a_n = 205
We know that
a_n = a + (n − 1) d
Therefore, 205 = 7 + (n − 1) 6
198 = (n − 1) 6
33 = (n − 1)
n = 34
Therefore, this given series has 34 terms in it.

(ii) Let there are n terms in this A.P.
a_n = 205
a_n = a + (n − 1) d-47 = 18 + (n – 1) (-5/2)-47 – 18 = (n – 1) (-5/2)
-65 = (n – 1)(-5/2)
(n – 1) = -130/-5
(n – 1) = 26
n = 27
Therefore, this given A.P. has 27 terms in it.

Question 6:
Check whether -150 is a term of the A.P. 11, 8, 5, 2,

Answer 6:
For this A.P.,
a = 11
d = a₂ − a₁ = 8 − 11 = −3
Let −150 be the n^th term of this A.P.
We know that,
a_n = a + (n − 1) d
-150 = 11 + (n – 1)(-3)
-150 = 11 – 3n + 3
-164 = -3n
n = 164/3
Clearly, n is not an integer.
Therefore, – 150 is not a term of this A.P.

Question 7:
Find the 31st term of an A.P. whose 11th term is 38 and the 16th term is 73.

Answer 7:
Given that,
a₁₁ = 38
a₁₆ = 73
We know that,
a_n = a + (n − 1) d
a₁₁ = a + (11 − 1) d
38 = a + 10d … (i)
Similarly,
a₁₆ = a + (16 − 1) d
73 = a + 15d … (ii)

On subtracting (i) from (ii), we get
35 = 5d
d = 7
From equation (i),
38 = a + 10 × (7)
38 − 70 = a
a = −32
a₃₁ = a + (31 − 1) d
= − 32 + 30 (7)
= − 32 + 210
= 178
Hence, 31^st term is 178.

Question 8:
An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.

Answer 8:
Given that,
a₃ = 12
a₅₀ = 106
We know that,
a_n = a + (n − 1) d
a₃ = a + (3 − 1) d
12 = a + 2d … (i)
Similarly, a₅₀ = a + (50 − 1) d
106 = a + 49d … (ii)
On subtracting (i) from (ii), we get
94 = 47d
d = 2
From equation (i), we get
12 = a + 2 (2)
a = 12 − 4 = 8
a₂₉ = a + (29 − 1) d
a₂₉ = 8 + (28)2
a₂₉ = 8 + 56 = 64
Therefore, 29^th term is 64.

Question 9:
If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively. Which term of this A.P. is zero.

Answer 9:
Given that,
3^rd term, a₃ = 4
and 9^th term, a₉ = −8

We know that,
a_n = a+(n−1)d

Therefore,

a₃ = a+(3−1)d
4 = a+2d ……………………………………… (i)
a₉ = a+(9−1)d
−8 = a+8d ………………………………………………… (ii)

On subtracting equation (i) from (ii), we will get here,

−12 = 6d
d = −2

From equation (i), we can write,

4 = a+2(−2)
4 = a−4
a = 8

Let n^th term of this A.P. be zero.
a_n = a+(n−1)d
0 = 8+(n−1)(−2)
0 = 8−2n+2
2n = 10
n = 5

Hence, 5^th term of this A.P. is 0.

Question 10:
If 17th term of an A.P. exceeds its 10th term by 7. Find the common difference.

Answer 7:
We know that, for an A.P series;

a_n = a+(n−1)d
a₁₇ = a+(17−1)d
a₁₇ = a +16d

In the same way,
a₁₀ = a+9d

As it is given in the question,
a₁₇ − a₁₀ = 7

Therefore,
(a +16d)−(a+9d) = 7
7d = 7
d = 1

Therefore, the common difference is 1

Question 11:
Which term of the A.P. 3, 15, 27, 39,.. will be 132 more than its 54th term?

Answer 11:
Given A.P. is 3, 15, 27, 39, …
a = 3
d = a₂ − a₁ = 15 − 3 = 12
a₅₄ = a + (54 − 1) d
= 3 + (53) (12)
= 3 + 636 = 639
132 + 639 = 771
We have to find the term of this A.P. which is 771.
Let n^th term be 771.
a_n = a + (n − 1) d
771 = 3 + (n − 1) 12
768 = (n − 1) 12
(n − 1) = 64
n = 65
Therefore, 65^th term was 132 more than 54^th term

Question 12:
Two APs have the same common difference. The difference between their 100th term is 100, what is the difference between their 1000th terms?

Answer 12:
Let the first term of these A.P.s be a₁ and a₂ respectively and the common difference of these A.P.s be d.
For first A.P.,
a₁₀₀ = a₁ + (100 − 1) d
= a₁ + 99d
a₁₀₀₀ = a₁ + (1000 − 1) d
a₁₀₀₀ = a₁ + 999d

For second A.P.,
a₁₀₀ = a₂ + (100 − 1) d
= a₂ + 99d
a₁₀₀₀ = a₂ + (1000 − 1) d
= a₂ + 999d
Given that, difference between
100^th term of these A.P.s = 100
Therefore, (a₁ + 99d) − (a₂ + 99d) = 100
a₁ − a₂ = 100 … (i)

Difference between 1000^th terms of these A.P.s
(a₁ + 999d) − (a₂ + 999d) = a₁ − a₂
From equation (i),
This difference, a₁ − a₂ = 100
Hence, the difference between 1000^th terms of these A.P. will be 100.

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Question 13:
How many three digit numbers are divisible by 7?

Answer 13:
First three-digit number that is divisible by 7 = 105
Next number = 105 + 7 = 112
Therefore, 105, 112, 119, …
All are three digit numbers which are divisible by 7 and thus, all these are terms of an A.P. having first term as 105 and common difference as 7.
The maximum possible three-digit number is 999. When we divide it by 7, the remainder will be 5. Clearly, 999 − 5 = 994 is the maximum possible three-digit number that is divisible by 7.
The series is as follows.
105, 112, 119, …, 994
Let 994 be the nth term of this A.P.

a = 105
d = 7
a_n = 994
n = ?
a_n = a + (n − 1) d
994 = 105 + (n − 1) 7
889 = (n − 1) 7
(n − 1) = 127
n = 128
Therefore, 128 three-digit numbers are divisible by 7.

Question 14:
How many multiples of 4 lie between 10 and 250?

Answer 14:
First multiple of 4 that is greater than 10 is 12. Next will be 16.
Therefore, 12, 16, 20, 24, …
All these are divisible by 4 and thus, all these are terms of an A.P. with first term as 12 and common difference as 4.
When we divide 250 by 4, the remainder will be 2. Therefore, 250 − 2 = 248 is divisible by 4.
The series is as follows.
12, 16, 20, 24, …, 248
Let 248 be the n^th term of this A.P.
a = 12
d = 4
a_n = 248
a_n = a + (n – 1) d
248 = 12 + (n – 1) × 4
236/4 = n – 1
59  = n – 1
n = 60
Therefore, there are 60 multiples of 4 between 10 and 250.

Question 15:
For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10, 17, … equal?

Answer 15:
63, 65, 67, …
a = 63
d = a₂ − a₁ = 65 − 63 = 2
n^th term of this A.P. = a_n = a + (n − 1) d
a_n= 63 + (n − 1) 2 = 63 + 2n − 2
a_n = 61 + 2n … (i)
3, 10, 17, …
a = 3
d = a₂ − a₁ = 10 − 3 = 7
n^th term of this A.P. = 3 + (n − 1) 7
a_n= 3 + 7n − 7
a_n = 7n − 4 … (ii)
It is given that, n^th term of these A.P.s are equal to each other.
Equating both these equations, we obtain
61 + 2n = 7n − 4
61 + 4 = 5n
5n = 65
n = 13
Therefore, 13^th terms of both these A.P.s are equal to each other.

Question 16:
Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12.

Answer 16:
a₃ = 16a + (3 − 1) d = 16
a + 2d = 16 … (i)
a₇ − a₅ = 12
[a+ (7 − 1) d] − [a + (5 − 1) d]= 12
(a + 6d) − (a + 4d) = 12
2d = 12
d = 6
From equation (i), we get,
a + 2 (6) = 16
a + 12 = 16
a = 4
Therefore, A.P. will be
4, 10, 16, 22, …

Question 17:
Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253.

Answer 17:
Given A.P. is
3, 8, 13, …, 253
Common difference for this A.P. is 5.
Therefore, this A.P. can be written in reverse order as
253, 248, 243, …, 13, 8, 5
For this A.P.,
a = 253
d = 248 − 253 = −5
n = 20
a₂₀ = a + (20 − 1) d
a₂₀ = 253 + (19) (−5)
a₂₀ = 253 − 95
a = 158
Therefore, 20^th term from the last term is 158.

Question 18:
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the A.P.

Answer 18:
We know that,
a_n = a + (n − 1) d
a₄ = a + (4 − 1) d
a₄ = a + 3d
Similarly,
a₈ = a + 7d
a₆ = a + 5d
a₁₀ = a + 9d
Given that, a₄ + a₈ = 24
a + 3d + a + 7d = 24
2a + 10d = 24
a + 5d = 12 … (i)
a₆ + a₁₀ = 44
a₆ + a₁₀ = 44
a + 5d + a + 9d = 44
2a + 14d = 44
a + 7d = 22 … (ii)
On subtracting equation (i) from (ii), we get,
2d = 22 − 12
2d = 10
d = 5
From equation (i), we get
a + 5d = 12
a + 5 (5) = 12
a + 25 = 12
a = −13
a₂ = a + d = − 13 + 5 = −8
a₃ = a₂ + d = − 8 + 5 = −3
Therefore, the first three terms of this A.P. are −13, −8, and −3.

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Question 19:
Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?

Answer 19:
It can be seen from the given question, that the incomes of Subba Rao increases every year by Rs.200 and hence, forms an AP.

Therefore, after 1995, the salaries of each year are;
5000, 5200, 5400, …

Here, first term, a = 5000
and common difference, d = 200

Let after nth year, his salary be Rs 7000.
Therefore, by the nth term formula of AP,

an = a+(n−1) d
7000 = 5000+(n−1)200
200(n−1)= 2000
(n−1) = 10
n = 11

Therefore, in 11th year, his salary will be Rs 7000.

Question 20:
Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her weekly savings become Rs 20.75, find n.

Answer 20:
Given that,
a = 5
d = 1.75
a_n = 20.75
n = ?
a_n = a + (n − 1) d
20.75 = 5 + (n – 1) × 1.75
15.75 = (n – 1) × 1.75
(n – 1) = 15.75/1.75 = 1575/175
= 63/7 = 9
n – 1 = 9
n = 10
Hence, n is 10.

Exercise 5.3 (Page 112)

Question 1:
Find the sum of the following APs.

(i) 2, 7, 12 ,…., to 10 terms.
(ii) − 37, − 33, − 29 ,…, to 12 terms
(iii) 0.6, 1.7, 2.8 ,…….., to 100 terms
(iv) 1/15, 1/12, 1/10, …… , to 11 terms

Answer 1:
(i) 2, 7, 12 ,…, to 10 terms
For this A.P.,
a = 2
d = a₂ − a₁ = 7 − 2 = 5
n = 10
We know that,
S_n = n/2 [2a + (n – 1) d]
S₁₀ = 10/2 [2(2) + (10 – 1) × 5]
= 5[4 + (9) × (5)]
= 5 × 49 = 245

(ii) −37, −33, −29 ,…, to 12 terms
For this A.P.,
a = −37
d = a₂ − a₁ = (−33) − (−37)
= − 33 + 37 = 4
n = 12
We know that,
S_n = n/2 [2a + (n – 1) d]
S₁₂ = 12/2 [2(-37) + (12 – 1) × 4]
= 6[-74 + 11 × 4]
= 6[-74 + 44]
= 6(-30) = -180

(iii) 0.6, 1.7, 2.8 ,…, to 100 terms
For this A.P.,
a = 0.6
d = a₂ − a₁ = 1.7 − 0.6 = 1.1
n = 100
We know that,
S_n = n/2 [2a + (n – 1) d]
S₁₂ = 50/2 [1.2 + (99) × 1.1]
= 50[1.2 + 108.9]
= 50[110.1]
= 5505

(iv) Given, 1/15, 1/12, 1/10, …… , to 11 terms
For this A.P.,
First term, a = 1/5
Common difference, d = a2 –a1 = (1/12)-(1/5) = 1/60
And number of terms n = 11
We know that, the formula for sum of nth term in AP series is,

S_n=n/2[2a=(n-1)d]
= 11/2(2/15 + 10/60)
= 11/2 (9/30)
= 33/20

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Question 2:
Find the sums given below
(i) 7 +10 1/2+ 14 + ……………… +84
(ii)+ 14 + ………… + 84
(ii) 34 + 32 + 30 + ……….. + 10
(iii) − 5 + (− 8) + (− 11) + ………… + (− 230)

Answer 2

Sn = n/2 (a + l) , l = 84
Sn = 23/2 (7+84)

Sn = (23×91/2) = 2093/2

(ii) 34 + 32 + 30 + ……….. + 10
For this A.P.,
a = 34
d = a₂ − a₁ = 32 − 34 = −2
l = 10
Let 10 be the n^th term of this A.P.
l = a + (n − 1) d
10 = 34 + (n − 1) (−2)
−24 = (n − 1) (−2)
12 = n − 1
n = 13
S_n = n/2 (a + l)
= 13/2 (34 + 10)
= (13×44/2) = 13 × 22
= 286

(iii) (−5) + (−8) + (−11) + ………… + (−230) For this A.P.,
a = −5
l = −230
d = a₂ − a₁ = (−8) − (−5)
= − 8 + 5 = −3
Let −230 be the n^th term of this A.P.
l = a + (n − 1)d
−230 = − 5 + (n − 1) (−3)
−225 = (n − 1) (−3)
(n − 1) = 75
n = 76
And,
S_n = n/2 (a + l)
= 76/2 [(-5) + (-230)]
= 38(-235)
= -8930

Question 3:
In an AP
(i) Given a = 5, d = 3, a_n = 50, find n and S_n.
(ii) Given a = 7, a₁₃ = 35, find d and S₁₃.
(iii) Given a₁₂ = 37, d = 3, find a and S₁₂.
(iv) Given a₃ = 15, S₁₀ = 125, find d and a₁₀.
(v) Given d = 5, S₉ = 75, find a and a₉.
(vi) Given a = 2, d = 8, S_n = 90, find n and a_n.
(vii) Given a = 8, a_n = 62, S_n = 210, find n and d.
(viii) Given a_n = 4, d = 2, S_n = − 14, find n and a.
(ix) Given a = 3, n = 8, S = 192, find d.
(x) Given l = 28, S = 144 and there are total 9 terms. Find a.

Answer 3:
(i) Given that, a = 5, d = 3, a_n = 50
As a_n = a + (n − 1)d,
⇒ 50 = 5 + (n – 1) × 3
⇒ 3(n – 1) = 45
⇒ n – 1 = 15
⇒ n = 16
Now, S_n = n/2 (a + a_n)
S_n = 16/2 (5 + 50) = 440

(ii) Given that, a = 7, a₁₃ = 35
As a_n = a + (n − 1)d, ⇒ 35 = 7 + (13 – 1)d
⇒ 12d = 28
⇒ d = 28/12 = 2.33
Now, S_n = n/2 (a + a_n)
S₁₃ = 13/2 (7 + 35) = 273

(iii)Given that, a₁₂ = 37, d = 3 As a_n = a + (n − 1)d,
⇒ a₁₂ = a + (12 − 1)3
⇒ 37 = a + 33
⇒ a = 4
S_n = n/2 (a + a_n)
S_n = 12/2 (4 + 37)
= 246

(iv) Given that, a₃ = 15, S₁₀ = 125
As a_n = a + (n − 1)d,
a₃ = a + (3 − 1)d
15 = a + 2d … (i)
S_n = n/2 [2a + (n – 1)d]
S₁₀ = 10/2 [2a + (10 – 1)d]
125 = 5(2a + 9d)
25 = 2a + 9d … (ii)
On multiplying equation (i) by (ii), we get
30 = 2a + 4d … (iii)
On subtracting equation (iii) from (ii), we get
−5 = 5d
d = −1

From equation (i),
15 = a + 2(−1)
15 = a − 2
a = 17
a₁₀ = a + (10 − 1)d
a₁₀ = 17 + (9) (−1)
a₁₀ = 17 − 9 = 8

(v) Given that, d = 5, S₉ = 75
As S_n = n/2 [2a + (n – 1)d]
S₉ = 9/2 [2a + (9 – 1)5]
25 = 3(a + 20)
25 = 3a + 60
3a = 25 − 60
a = -35/3
a_n = a + (n − 1)d
a₉ = a + (9 − 1) (5)
= -35/3 + 8(5)
= -35/3 + 40
= (35+120/3) = 85/3

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(vi) Given that, a = 2, d = 8, S_n = 90
As S_n = n/2 [2a + (n – 1)d]
90 = n/2 [2a + (n – 1)d]
⇒ 180 = n(4 + 8n – 8) = n(8n – 4) = 8n² – 4n
⇒ 8n² – 4n – 180 = 0
⇒ 2n² – n – 45 = 0
⇒ 2n² – 10n + 9n – 45 = 0
⇒ 2n(n -5) + 9(n – 5) = 0
⇒ (2n – 9)(2n + 9) = 0
So, n = 5 (as it is positive integer)
∴ a₅ = 8 + 5 × 4 = 34

(vii) Given that, a = 8, a_n = 62, S_n = 210
As S_n = n/2 (a + a_n)
210 = n/2 (8 + 62)
⇒ 35n = 210
⇒ n = 210/35 = 6
Now, 62 = 8 + 5d
⇒ 5d = 62 – 8 = 54
⇒ d = 54/5 = 10.8

(viii) Given that, a_n = 4, d = 2, S_n = −14
a_n = a + (n − 1)d
4 = a + (n − 1)2
4 = a + 2n − 2
a + 2n = 6
a = 6 − 2n … (i)
S_n = n/2 (a + a_n)
-14 = n/2 (a + 4)
−28 = n (a + 4)
−28 = n (6 − 2n + 4) {From equation (i)}
−28 = n (− 2n + 10)

−28 = − 2n² + 10n
2n² − 10n − 28 = 0
n² − 5n −14 = 0
n² − 7n + 2n − 14 = 0
n (n − 7) + 2(n − 7) = 0
(n − 7) (n + 2) = 0
Either n − 7 = 0 or n + 2 = 0
n = 7 or n = −2
However, n can neither be negative nor fractional.
Therefore, n = 7
From equation (i), we get

a = 6 − 2n
a = 6 − 2(7)
= 6 − 14
= −8

(ix) Given that, a = 3, n = 8, S = 192
As S_n = n/2 [2a + (n – 1)d]
192 = 8/2 [2 × 3 + (8 – 1)d]
192 = 4 [6 + 7d]
48 = 6 + 7d
42 = 7d
d = 6

(x) Given that, l = 28, S = 144 and there are total of 9 terms.
S_n = n/2 (a + l)
144 = 9/2 (a + 28)
(16) × (2) = a + 28
32 = a + 28
a = 4

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Question 4:
How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?

Answer 4:
Let there be n terms of this A.P.
For this A.P., a = 9
d = a₂ − a₁ = 17 − 9 = 8
As S_n = n/2 [2a + (n – 1)d]
636 = n/2 [2 × a + (8 – 1) × 8]
636 = n/2 [18 + (n– 1) × 8]
636 = n [9 + 4n − 4]
636 = n (4n + 5)
4n² + 5n − 636 = 0
4n² + 53n − 48n − 636 = 0
n (4n + 53) − 12 (4n + 53) = 0
(4n + 53) (n − 12) = 0
Either 4n + 53 = 0 or n − 12 = 0
n = (-53/4) or n = 12
n cannot be (-53/4). As the number of terms can neither be negative nor fractional, therefore, n = 12 only.

Question 5:
The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Answer 5:
Given that,
first term, a = 5
last term, l = 45

Sum of the AP, Sn = 400
As we know, the sum of AP formula is;

Sn = n/2 (a+l)
400 = n/2(5+45)
400 = n/2(50)

Number of terms, n =16
As we know, the last term of AP series can be written as;

l = a+(n −1)d
45 = 5 +(16 −1)d
40 = 15d

Common difference, d = 40/15 = 8/3

Question 6:
The first and the last term of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

Answer 6:
Given that,
a = 17
l = 350
d = 9
Let there be n terms in the A.P.
l = a + (n − 1) d
350 = 17 + (n − 1)9
333 = (n − 1)9
(n − 1) = 37
n = 38
S_n = n/2 (a + l)
S₃₈ = 13/2 (17 + 350)
= 19 × 367
= 6973
Thus, this A.P. contains 38 terms and the sum of the terms of this A.P. is 6973.

Question 7:
Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

Answer 7:
d = 7
a₂₂ = 149
S₂₂ = ?
a_n = a + (n − 1)d
a₂₂ = a + (22 − 1)d
149 = a + 21 × 7
149 = a + 147
a = 2
S_n = n/2 (a + a_n)
= 22/2 (2 + 149)
= 11 × 151
= 1661

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Question 8:
Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

Answer 8:
Given that,
a₂ = 14
a₃ = 18
d = a₃ − a₂ = 18 − 14 = 4
a₂ = a + d
14 = a + 4
a = 10
S_n = n/2 [2a + (n – 1)d]
S₅₁ = 51/2 [2 × 10 + (51 – 1) × 4]
= 51/2 [2 + (20) × 4]
= 51×220/2
= 51 × 110
= 5610

Question 9:
If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Answer 9:
Given that,S₇ = 49
S₁₇ = 289
S₇  = 7/2 [2a + (n – 1)d]
S₇ = 7/2 [2a + (7 – 1)d]
49 = 7/2 [2a + 16d]
7 = (a + 3d)
a + 3d = 7 … (i)
Similarly,
S₁₇ = 17/2 [2a + (17 – 1)d]
289 = 17/2 (2a + 16d)
17 = (a + 8d)
a + 8d = 17 … (ii)
Subtracting equation (i) from equation (ii),
5d = 10
d = 2
From equation (i),
a + 3(2) = 7
a + 6 = 7
a = 1
S_n = n/2 [2a + (n – 1)d]
= n/2 [2(1) + (n – 1) × 2]
= n/2 (2 + 2n – 2)
= n/2 (2n)
= n²

Question 10:
Show that a₁, a₂ … , a_n , … form an AP where a_n is defined as below
(i) a_n = 3+4n
(ii) a_n = 9−5n
Also find the sum of the first 15 terms in each case.

Answer 10:

(i) a_n = 3 + 4n
a₁ = 3 + 4(1) = 7
a₂ = 3 + 4(2) = 3 + 8 = 11
a₃ = 3 + 4(3) = 3 + 12 = 15
a₄ = 3 + 4(4) = 3 + 16 = 19
It can be observed that
a₂ − a₁ = 11 − 7 = 4
a₃ − a₂ = 15 − 11 = 4
a₄ − a₃ = 19 − 15 = 4
i.e., a_{k + 1} − a_k is same every time. Therefore, this is an AP with common difference as 4 and first term as 7.
S_n = n/2 [2a + (n – 1)d]
S₁₅ = 15/2 [2(7) + (15 – 1) × 4]
= 15/2 [(14) + 56]
= 15/2 (70)
= 15 × 35
= 525

(ii) a_n = 9 − 5n
a₁ = 9 − 5 × 1 = 9 − 5 = 4
a₂ = 9 − 5 × 2 = 9 − 10 = −1
a₃ = 9 − 5 × 3 = 9 − 15 = −6
a₄ = 9 − 5 × 4 = 9 − 20 = −11
It can be observed that
a₂ − a₁ = − 1 − 4 = −5
a₃ − a₂ = − 6 − (−1) = −5
a₄ − a₃ = − 11 − (−6) = −5
i.e., a_{k + 1} − a_k is same every time. Therefore, this is an A.P. with common difference as −5 and first term as 4.
S_n = n/2 [2a + (n – 1)d]
S₁₅ = 15/2 [2(4) + (15 – 1) (-5)]
= 15/2 [8 + 14(-5)]
= 15/2 (8 – 70)
= 15/2 (-62)
= 15(-31)
= -465

Question 11:
If the sum of the first n terms of an AP is 4n − n², what is the first term (that is S₁)? What is the sum of first two terms? What is the second term? Similarly find the 3^rd, the10^th and the n^th terms.

Answer 11:
Given that,
S_n = 4n − n²
First term, a = S₁ = 4(1) − (1)² = 4 − 1 = 3
Sum of first two terms = S₂
= 4(2) − (2)² = 8 − 4 = 4
Second term, a₂ = S₂ − S₁ = 4 − 3 = 1
d = a₂ − a = 1 − 3 = −2
a_n = a + (n − 1)d
= 3 + (n − 1) (−2)
= 3 − 2n + 2
= 5 − 2n
Therefore, a₃ = 5 − 2(3) = 5 − 6 = −1
a₁₀ = 5 − 2(10) = 5 − 20 = −15
Hence, the sum of first two terms is 4. The second term is 1. 3^rd, 10^th, and n^th terms are −1, −15, and 5 − 2n respectively.

Question 12:
Find the sum of first 40 positive integers divisible by 6.

Answer 12:
The positive integers that are divisible by 6 are
6, 12, 18, 24 …
It can be observed that these are making an A.P. whose first term is 6 and common difference is 6.
a = 6
d = 6
S₄₀ = ?
S_n = n/2 [2a + (n – 1)d]
S₄₀ = 40/2 [2(6) + (40 – 1) 6]
= 20[12 + (39) (6)]
= 20(12 + 234)
= 20 × 246
= 4920

Question 13:
Find the sum of first 15 multiples of 8.

Answer 13:
The multiples of 8 are 8, 16, 24, 32…
The series is in the form of AP, having first term as 8 and common difference as 8.
Therefore, a = 8

d = 8
S₁₅ = ?

By the formula of sum of nth term, we know,
S_n = n/2 [2a+(n-1)d]
S₁₅ = 15/2 [2(8) + (15-1)8]
= 15/2[6 +(14)(8)]
= 15/2[16 +112]
= 15(128)/2
= 15 × 64
= 960

Question 14:
Find the sum of the odd numbers between 0 and 50.

Answer 14:
The odd numbers between 0 and 50 are
1, 3, 5, 7, 9 … 49
Therefore, it can be observed that these odd numbers are in an A.P.
a = 1
d = 2
l = 49
l = a + (n − 1) d
49 = 1 + (n − 1)2
48 = 2(n − 1)
n − 1 = 24
n = 25
S_n = n/2 (a + l)

S₂₅ = 25/2 (1 + 49)
= 25(50)/2
=(25)(25)
= 625

Question 15:
A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days.

Answer 15:
It can be observed that these penalties are in an A.P. having first term as 200 and common difference as 50.
a = 200
d = 50
Penalty that has to be paid if he has delayed the work by 30 days = S₃₀
= 30/2 [2(200) + (30 – 1) 50]

= 15 [400 + 1450]
= 15 (1850)
= 27750
Therefore, the contractor has to pay Rs 27750 as penalty.

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Question 16:
A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.

Answer 16:
Let the cost of 1^st prize be P.
Cost of 2^nd prize = P − 20
And cost of 3^rd prize = P − 40
It can be observed that the cost of these prizes are in an A.P. having common difference as −20 and first term as P.
a = P
d = −20
Given that, S₇ = 700
7/2 [2a + (7 – 1)d] = 700

a + 3(−20) = 100
a − 60 = 100
a = 160
Therefore, the value of each of the prizes was Rs 160, Rs 140, Rs 120, Rs 100, Rs 80, Rs 60, and Rs 40.

Question 17:
In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of class I will plant 1 tree, a section of class II will plant 2 trees and so on till class XII. There are three sections of each class. How many trees will be planted by the students?

Answer 17:
It can be observed that the number of trees planted by the students is in an AP.
1, 2, 3, 4, 5………………..12
First term, a = 1
Common difference, d = 2 − 1 = 1
S_n = n/2 [2a + (n – 1)d]
S₁₂ = 12/2 [2(1) + (12 – 1)(1)]
= 6 (2 + 11)
= 6 (13)
= 78
Therefore, number of trees planted by 1 section of the classes = 78

Question 18:
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π = 22/7)

Answer 18:
perimeter of semi-circle = πr
_P1 = π(0.5) = π/2 cm
_P2 = π(1) = π cm
_P3 = π(1.5) = 3π/2 cm
_P1, P₂, P₃ are the lengths of the semi-circles
π/2, π, 3π/2, 2π, ….
P1= π/2 cm
_P2 = π cm
d = P2- P1 = π – π/2 = π/2
First term = P1 = a = π/2 cm
S_n = n/2 [2a + (n – 1)d]

Therefor, Sum of the length of 13 consecutive circles
S₁₃ = 13/2 [2(π/2) + (13 – 1)π/2]
=  13/2 [π + 6π]
=13/2 (7π)  = 13/2 × 7 × 22/7
= 143 cm

Question 19:
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?

Answer 19:

It can be observed that the numbers of logs in rows are in an A.P.
20, 19, 18…
For this A.P.,
a = 20
d = a₂ − a₁ = 19 − 20 = −1
Let a total of 200 logs be placed in n rows.
S_n = 200
S_n = n/2 [2a + (n – 1)d]
S₁₂ = 12/2 [2(20) + (n – 1)(-1)]
400 = n (40 − n + 1)
400 = n (41 − n)
400 = 41n − n²
n² − 41n + 400 = 0
n² − 16n − 25n + 400 = 0
n (n − 16) −25 (n − 16) = 0
(n − 16) (n − 25) = 0
Either (n − 16) = 0 or n − 25 = 0
n = 16 or n = 25
a_n = a + (n − 1)d
a₁₆ = 20 + (16 − 1) (−1)
a₁₆ = 20 − 15
a₁₆ = 5

Similarly,
a₂₅ = 20 + (25 − 1) (−1)
a₂₅ = 20 − 24
= −4
Clearly, the number of logs in 16^th row is 5. However, the number of logs in 25^th row is negative, which is not possible.
Therefore, 200 logs can be placed in 16 rows and the number of logs in the 16^th row is 5.

Question 20:
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.

A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2×5+2×(5+3)]

Answer 20:
The distances of potatoes from the bucket are 5, 8, 11, 14…
Distance run by the competitor for collecting these potatoes are two times of the distance at which the potatoes have been kept because first she has to first pick the potato and again return back to the same place in order to start picking the second potato.. Therefore, distances to be run are
10, 16, 22, 28, 34,……….
a = 10
d = 16 − 10 = 6
S₁₀ =?
S₁₀ = 10/2 [2(20) + (n – 1)(-1)]
= 5[20 + 54]
= 5 (74)
= 370
Therefore, the competitor will run a total distance of 370 m.

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