# NCERT Solutions class 10 Maths Ch- 5 Arithmetic Progressions

## EX - 5.1

1. In which of the following situations, does the list of numbers involved make as arithmetic progression and why?

(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.

(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.

Solution:

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) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.

Solution:

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) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.

Solution:

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

Solution:

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.

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

Solutions:

(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 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.

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 …

Solutions

(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

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) aa2a3a4 …
(xii) √2, √8, √18, √32 …
(xiii) √3, √6, √9, √12 …
(xiv) 12, 32, 52, 72 …
(xv) 12, 52, 72, 73 …

Solution

(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, an+1 – a 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, an+1 – an 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, an+1 – an 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, an+1 – an 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, an+1 – an 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, an+1 – an or the common difference is not same every time.

Therefore, and the given series doesn’t forms a A.P.

(vii) 0, -4, -8, -12 …

Here,

a2 – a1 = (-4)-0 = -4

a3 – a2 = (-8)-(-4) = -4

a4 – a3 = (-12)-(-8) = -4

Since, an+1 – an or the common difference is same every time.

Therefore, d = -4 and 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, an+1 – an 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, an+1 – an or the common difference is not same every time.

Therefore, and the given series doesn’t forms a A.P.

(x) a, 2a, 3a, 4a …

Here,

a2 – a1 = 2aa

a3 – a2 = 3a-2a = a

a4 – a3 = 4a-3a = a

Since, an+1 – an 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) aa2a3a4 …

Here,

a2 – a1 = a2a = a(a-1)

a3 – a2 = a aa2(a-1)

a4 – a3 = a4 – aa3(a-1)

Since, an+1 – an 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, an+1 – an 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, an+1 – an or the common difference is not same every time.

Therefore, the given series doesn’t forms a A.P.

(xiv) 12, 32, 52, 72 …

Or, 1, 9, 25, 49 …..

Here,

a2 − a1 = 9−1 = 8

a3 − a= 25−9 = 16

a4 − a3 = 49−25 = 24

Since, an+1 – an or the common difference is not same every time.

Therefore, the given series doesn’t forms a A.P.

(xv) 12, 52, 72, 73 …

Or 1, 25, 49, 73 …

Here,

a2 − a1 = 25−1 = 24

a3 − a= 49−25 = 24

a4 − a3 = 73−49 = 24

Since, an+1 – an 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

a= 121+24 = 145

## EX - 5.2

Solutions:

(i) Given,

First term, a = 7

Common difference, d = 3

Number of terms, n = 8,

We have to find the nth term, an = ?

As we know, for an A.P.,

an = a+(n−1)d

Putting the values,

=> 7+(8 −1) 3

=> 7+(7) 3

=> 7+21 = 28

Hence, an = 28

(ii) Given,

First term, a = -18

Common difference, d = ?

Number of terms, n = 10

Nth term, an = 0

As we know, for an A.P.,

an = a+(n−1)d

Putting the values,

0 = − 18 +(10−1)d

18 = 9d

d = 18/9 = 2

Hence, common difference, = 2

(iii) Given,

First term, a = ?

Common difference, d = -3

Number of terms, n = 18

Nth term, an = -5

As we know, for an A.P.,

an = a+(n−1)d

Putting the values,

−5 = a+(18−1) (−3)

−5 = a+(17) (−3)

−5 = a−51

a = 51−5 = 46

Hence, a = 46

(iv) Given,

First term, a = -18.9

Common difference, d = 2.5

Number of terms, n = ?

Nth term, an = 3.6

As we know, for an A.P.,

an = a +(n −1)d

Putting the values,

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) Given,

First term, a = 3.5

Common difference, d = 0

Number of terms, n = 105

Nth term, an = ?

As we know, for an A.P.,

an = a+(n −1)d

Putting the values,

an = 3.5+(105−1) 0

an = 3.5+104×0

an = 3.5

Hence, an = 3.5

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)20

Solutions:

(i) Given here,

A.P. = 10, 7, 4, …

Therefore, we can find,

First term, a = 10

Common difference, d = a2 − a= 7−10 = −3

As we know, for an A.P.,

an = a +(n−1)d

Putting the values;

a30 = 10+(30−1)(−3)

a30 = 10+(29)(−3)

a30 = 10−87 = −77

Hence, the correct answer is option C.

(ii) Given here,

A.P. = -3, -1/2, ,2 …

Therefore, we can find,

First term a = – 3

Common difference, d = a2 − a1 = (-1/2) -(-3)

⇒(-1/2) + 3 = 5/2

As we know, for an A.P.,

an = a+(n−1)d

Putting the values;

a11 = 3+(11-1)(5/2)

a11 = 3+(10)(5/2)

a11 = -3+25

a11 = 22

Hence, the answer is option B.

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

(i) 2, __,26
(ii) __, 13, __, 3
(iii) 5, __, __, 19/2

(iv) -4, __, __, __, __, 6
(v) __, 38, __, __, __, -22

Solutions:

(i) For the given A.P., 2, __,26

The first and third term are;

a = 2

a3 = 26

As we know, for an A.P.,

an = a+(n −1)d

Therefore, putting the values here,

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 the given A.P., __, 13, __, 3

a2 = 13 and

a4 = 3

As we know, for an A.P.,

an = a+(n−1) d

Therefore, putting the values here,

a2 = a +(2-1)d

13 = a+d ………………. (i)

a4 = a+(4-1)d

3 = a+3d ………….. (ii)

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

– 10 = 2d

d = – 5

From equation (i), putting the value of d,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 the given A.P.,5, __, __,19/2

= 5 and

a4 = 19/2

As we know, for an A.P.,

an = a+(n−1)d

Therefore, putting the values here,

a4 = a+(4-1)d

19/2 = 5+3d

(19/2) – 5 = 3d

3d = 9/2

d = 3/2

a2 = a+(2-1)d

a2 = 5+3/2

a2 = 13/2

a3 = a+(3-1)d

a3 = 5+2×3/2

a3 = 8

Therefore, the missing terms are 13/2 and 8 respectively.

(iv) For the given A.P., -4, __, __, __, __, 6

a = −4 and

a6 = 6

As we know, for an A.P.,

an = a +(n−1) d

Therefore, putting the values here,

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 the given A.P., __, 38, __, __, __, -22

a2 = 38

a6 = −22

As we know, for an A.P.,

an = a+(n −1)d

Therefore, putting the values here,

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 = + 2= 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.

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

Solutions:

Given the A.P. series as3, 8, 13, 18, …

First term, a = 3

Common difference, d = a2 − a1 = 8 − 3 = 5

Let the nth term of given A.P. be 78. Now as we know,

an = a+(n−1)d

Therefore,

78 = 3+(n −1)5

75 = (n−1)5

(n−1) = 15

n = 16

Hence, 16th term of this A.P. is 78.

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

(i) 7, 13, 19, …, 205

(ii) 18,, 13… -47

Solutions:

(i) Given, 7, 13, 19, …, 205 is the A.P

Therefore

First term, a = 7

Common difference, d = a2 − a1 = 13 − 7 = 6

Let there are n terms in this A.P.

an = 205

As we know, for an A.P.,

an = a + (n − 1) d

Therefore, 205 = 7 + (− 1) 6

198 = (n − 1) 6

33 = (n − 1)

n = 34

Therefore, this given series has 34 terms in it.

(ii) Given, 18,, 13… -47 is the A.P.

First term, a = 18

Common difference, d = a2-a= 15 1/2

d = (31-36)/2 = -5/2

Let there are n terms in this A.P.

an = 205

As we know, for an A.P.,

an = 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.

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

Solution:

For the given series, A.P. 11, 8, 5, 2..

First term, a = 11

Common difference, d = a2a1 = 8−11 = −3

Let −150 be the nth term of this A.P.

As we know, for an A.P.,

an = a+(n−1)d

-150 = 11+(n -1)(-3)

-150 = 11-3n +3

-164 = -3n

n = 164/3

Clearly, n is not an integer but a fraction.

Therefore, – 150 is not a term of this A.P.

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

Solution:

Given that,

11th term, a11 = 38

and 16th term, a16 = 73

We know that,

an = a+(n−1)d

a11 = a+(11−1)d

38 = a+10d ………………………………. (i)

In the same way,

a16 = a +(16−1)d

73 = a+15d ………………………………………… (ii)

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

35 = 5d

d = 7

From equation (i), we can write,

38 = a+10×(7)

38 − 70 = a

a = −32

a31 = a +(31−1) d

= − 32 + 30 (7)

= − 32 + 210

= 178

Hence, 31st term is 178.

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

Solution: Given that,

3rd term, a3 = 12

50th term, a50 = 106

We know that,

an = a+(n−1)d

a3 = a+(3−1)d

12 = a+2d ……………………………. (i)

In the same way,

a50 a+(50−1)d

106 = a+49d …………………………. (ii)

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

94 = 47d

d = 2 = common difference

From equation (i), we can write now,

12 = a+2(2)

a = 12−4 = 8

a29 = a+(29−1) d

a29 = 8+(28)2

a29 = 8+56 = 64

Therefore, 29th term is 64.

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.

Solution:

Given that,

3rd term, a3 = 4

and 9th term, a9 = −8

We know that,

an = 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 nth term of this A.P. be zero.

aa+(n−1)d

0 = 8+(n−1)(−2)

0 = 8−2n+2

2= 10

n = 5

Hence, 5th term of this A.P. is 0.

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

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

an = 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.

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

Solution:Given A.P. is 3, 15, 27, 39, …

first term, a = 3

common difference, d = a2 − a1 = 15 − 3 = 12

We know that,

an = a+(n−1)d

Therefore,

a54 = a+(54−1)d

⇒3+(53)(12)

⇒3+636 = 639

a54 = 639

We have to find the term of this A.P. which is 132 more than a54, i.e.771.

Let nth term be 771.

an = a+(n−1)d

771 = 3+(n −1)12

768 = (n−1)12

(n −1) = 64

n = 65

Therefore, 65th term was 132 more than 54th term.

Or another method is;

Let nth term be 132 more than 54th term.

n = 54 + 132/2

= 54 + 11 = 65th term

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

Solution:Let,

The first term of two APs be a1 and a2 respectively

And the common difference of these APs be d.

For the first A.P.,we know,

an = a+(n−1)d

Therefore,

a100 = a1+(100−1)d

a1 + 99d

a1000 = a1+(1000−1)d

a1000 = a1+999d

For second A.P., we know,

an = a+(n−1)d

Therefore,

a100 = a2+(100−1)d

a2+99d

a1000 = a2+(1000−1)d

a2+999d

Given that, difference between 100th term of the two APs = 100

Therefore, (a1+99d) − (a2+99d) = 100

a1a2 = 100……………………………………………………………….. (i)

Difference between 1000th terms of the two APs

(a1+999d) − (a2+999d) = a1a2

From equation (i),

This difference, a1a= 100

Hence, the difference between 1000th terms of the two A.P. will be 100.

13. How many three digit numbers are divisible by 7?

Solution:First three-digit number that is divisible by 7 are;

First number = 105

Second number = 105+7 = 112

Third number = 112+7 =119

Therefore, 105, 112, 119, …

All are three digit numbers are divisible by 7 and thus, all these are terms of an A.P. having first term as 105 and common difference as 7.

As we know, the largest possible three-digit number is 999.

When we divide 999 by 7, the remainder will be 5.

Therefore, 999-5 = 994 is the maximum possible three-digit number that is divisible by 7.

Now the series is as follows.

105, 112, 119, …, 994

Let 994 be the nth term of this A.P.

first term, a = 105

common difference, d = 7

an = 994

n = ?

As we know,

an = 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.

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

Solution: The first multiple of 4 that is greater than 10 is 12.

Next multiple will be 16.

Therefore, the series formed as;

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, now;

12, 16, 20, 24, …, 248

Let 248 be the nth term of this A.P.

first term, a = 12

common difference, d = 4

an = 248

As we know,

an = 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.

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

Solution: Given two APs as; 63, 65, 67,… and 3, 10, 17,….

Taking first AP,

63, 65, 67, …

First term, a = 63

Common difference, d = a2−a1 = 65−63 = 2

We know, nth term of this A.P. = an = a+(n−1)d

an= 63+(n−1)2 = 63+2n−2

an = 61+2n ………………………………………. (i)

Taking second AP,

3, 10, 17, …

First term, a = 3

Common difference, d = a2 − a1 = 10 − 3 = 7

We know that,

nth term of this A.P. = 3+(n−1)7

an = 3+7n−7

an = 7n−4 ……………………………….. (ii)

Given, nth term of these A.P.s are equal to each other.

Equating both these equations, we get,

61+2n = 7n−4

61+4 = 5n

5n = 65

n = 13

Therefore, 13th terms of both these A.P.s are equal to each other.

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

Solutions: Given,

Third term, a3 = 16

As we know,

a +(3−1)d = 16

a+2d = 16 ………………………………………. (i)

It is given that, 7th term exceeds the 5th term by 12.

a7 − a5 = 12

[a+(7−1)d]−[+(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 be4, 10, 16, 22, …

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

Solutions: Given A.P. is3, 8, 13, …, 253

Common difference, d= 5.

Therefore, we can write the given AP in reverse order as;

253, 248, 243, …, 13, 8, 5

Now for the new AP,

first term, a = 253

and common difference, d = 248 − 253 = −5

n = 20

Therefore, using nth term formula, we get,

a20 = a+(20−1)d

a20 = 253+(19)(−5)

a20 = 253−95

a = 158

Therefore, 20th term from the last term of the AP 3, 8, 13, …, 253.is 158.

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.

Solutions: We know that, the nth term of the AP is;

an = a+(n−1)d

a4 = a+(4−1)d

a4 = a+3d

In the same way, we can write,

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

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.

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?

Solution: 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.

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.

Solution: Given that, Ramkali saved Rs.5 in first week and then started saving each week by Rs.1.75.

Hence,

First term, a = 5

and common difference, d = 1.75

Also given,

a= 20.75

Find, n = ?

As we know, by the nth term formula,

an = a+(n−1)d

Therefore,

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.

## Ex 5.3

Solutions:

#### Sn = n/2 (a + l) 144 = 9/2(a+28) (16)×(2) = a+28 32 = a+28a = 4

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

Solutions:

#### Hence, ak + 1 − ak is same every time. Therefore, this is an A.P. with common difference as −5 and first term as 4. Now, we know, the sum of nth term is;Sn = 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

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 3rd, the10th and the nth terms.

#### P1= π/2 cmP2 = π cmCommon difference, d = P2- P1 = π – π/2 = π/2 First term = P1= a = π/2 cm By the sum of n term formula, we know,Sn = n/2 [2a + (n – 1)d] Therefor, Sum of the length of 13 consecutive circles is;S13 = 13/2 [2(π/2) + (13 – 1)π/2] =  13/2 [π + 6π]=13/2 (7π) = 13/2 × 7 × 22/7 = 143 cm

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?

#### Let a total of 200 logs be placed in n rows.Thus, Sn = 200 By the sum of nth term formula,Sn = 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−n2n2−41n + 400 = 0n2−16n−25n+400 = 0n(n −16)−25(n −16) = 0 (n −16)(n −25) = 0 Either (n −16) = 0 or n−25 = 0n = 16 or n = 25 By the nth term formula,an = a+(n−1)da16 = 20+(16−1)(−1)a16 = 20−15a16 = 5 Similarly, the 25th term could be written as;a25 = 20+(25−1)(−1)a25 = 20−24= −4 It can be seen, the number of logs in 16th row is 5 as the numbers cannot be negative. Therefore, 200 logs can be placed in 16 rows and the number of logs in the 16th row is 5.

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)]

## Ex 5.4

1. Which term of the AP : 121, 117, 113, . . ., is its first negative term? [Hint : Find n for an < 0]

#### Therefore, an = 121+(n−1)(-4) = 121-4n+4 =125-4n To find the first negative term of the series, an < 0 Therefore, 125-4n < 0 125 < 4n n>125/4 n>31.25Therefore, the first negative term of the series is 32nd term.

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

#### = (x-1)/2[2.1+(x-1-1)1] = (x-1)/2 [2+x-2] = x(x-1)/2 ………………………………………(i) By the given condition, we can write, S49 – Sx = {49/2[2.1+(49-1)1]}–{x/2[2.1+(x-1)1]} = 25(49) – x(x + 1)/2 ………………………………….(ii) As per the given condition, eq.(i) and eq(ii) are equal to each other; Therefore, x(x-1)/2 = 25(49) – x(x-1)/2 x = ±35 As we know, the number of house cannot be an a negative number. Hence, the value of x is 35.

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1 4 m and a tread of 1 2 m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = ¼ ×1/2 ×50 m3.]