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```Chapter 9: Arithmetic Progressions Exercise 9.4

Question: 1

If 12th of an A.P is 82 and 18th term is 124. Then find out the 24th term.

Solution:

Given: a12 = 82 and a18 = 124

we know :

an = a + (n - 1) c. d

⟹ a12 = a + (12 - 1) c. d

⟹ 82 = a + 11 c. d  ... (1)

⟹ 124 = a + (18 - 1) c. d

⟹ 124 = a + 17 c. d  .... (2)

Subtracting (2) from (1)

⟹ (a + 17 c. d) - (a + 11c.d) = 124 - 82

⟹ a +17c.d - a - 11c.d = 42

⟹ 6 c.d = 42

⟹ c. d = 7

Here we have,

Common Difference (c. d) = 7 putting c. d = 7 in equation (1), we get

⟹ a + 11 × 7 = 82

⟹ a = 82 - 77

⟹ a = 5

Now, we have

First Term (a) = 5 we have to find 24th term

a24 = a + (24 - 1) c. d

= 5 + 23 × 7

= 5 + 161 = 166

Question: 2

In an A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Solution:

Given

24th term is twice the 10th term a24 = 2× a10       . . . . . . (1)

Let, first term be a and common difference be

d we know, nth term is  an = a + (n - 1)d

from equation (1), we have

a + (24 - 1)d = 2(a + (10 - 1)d)

⟹ a + 23d = 2 (a + 9d)

⟹ a + 23 d = 2a + 18d

⟹ (23 - 18)d = a

⟹ a = 5d

We have to prove that, 72nd term is twice the 34th term

⟹ a72 = 2 × a34

⟹ a + (72 - 1)d = 2 [a + (34 - 1)d]

⟹ a + 71d = 2(a + 33d)

⟹ a + 71d = 2a + 66d

Putting the value of a = 5 in the above equation,

⟹ 5d + 71d = 2 (5d) + 66d

⟹ 76d = 76d Hence it is proved…

Question: 3

If the (m + 1)th term of an A.P. is twice the (n + 1)th term of the A.P. Then prove that: (3m + 1)th is twice the (m + n + 1)th term.

Solution:

From the question, we have

a(m+1) = 2 a(n+1)

Let, First term = a and Common Difference = d

⟹ a + (m + 1 - 1)d = 2[a+(n + 1 – 1)d]

⟹ a + md = 2a + 2nd

⟹ a = md - 2nd

⟹ a = (m - 2n)d   .... (2)

We have to prove, a(3m+1) = 2a(m+n+1)

⟹ a + (3m + 1 - 1)d = 2 [a + (m + n + 1 -1)d]

⟹ a + 3md = 2a + 2(m + n)d putting the value of a = (m - 2n)d, from equation (1)

⟹ (m - 2n)d + 3md = 2[(m - 2n)d] + 2( m + n)d

⟹ m - 2n + 3m = 2m - 4n + 2m + 2n

⟹ 4m - 2n = 4m - 2n Hence it is proved…

Question: 4

If the nth term of the A.P. 9, 7, 5, … is same as the nth term of the A.P. 15, 12 , 9, … find n.

Solution:

We have here, First sequence is 9, 7, 5, … First term (a) = 9, Common Difference (c. d) = 9 - 7 = – 2 nth term

= a + (n-1) c. d

⟹ an = 9 + (n - 1)(– 2)

= 9 - 2n + 2 = 11 - 2n

Second sequence is 15, 12, 9, … here,

First term (a) = 15

Common Difference (c. d) = 12 -15 = -3 nth term = a + (n -1)d

⟹ a’n = 15 + (n - 1)(-3)

= 15 - 3n + 3 = 18 - 3n

We are given in the question that the nth term of both the A.P.s are same, So, we can write it as

a­n = a’n

⟹ 11 - 2n = 18 - 3n

⟹ n = 7

So, the 7th term of both the A.P.s will be equal.

Question: 5

Find the 13th term from the end in the following A.P.
(i)  4, 9, 14, … , 254.

(ii) 3, 5, 7, 9, …, 201.

(iii) 1, 4, 7, 10, … , 88.

Solution:

(i) We have,

First term (a) = 4 and common difference (c. d)

= 9 - 4 = 5

last term here (l) = 254 nth term from the end is :  l - (n - 1)d we have to find 13th term from end then :

l - 12d = 254 - 12 × 5

= 254 - 60 = 194

(ii). 3, 5, 7, 9, …, 201.

Solution:

we have, First term (a) = 3 and common difference (c. d)

= 5 - 3 = 2

last term here (l) = 201 nth term from the end is :

l - (n-1)d we have to find 13th term from end then :

l - 12d = 201 - 12 × 2

= 201 - 24 = 177
(iii). 1, 4, 7, 10, … , 88.

We have, First term (a) = 1 and common difference (c. d)

= 4 -1 = 3 last term here (l) = 88 nth term from the end is:

l - (n-1)d we have to find 13th term from end then:

l - 12d = 88 - 12 × 3

= 88 - 36 = 52

Question: 6

The 4th term of an A.P. is three times the first term and the 7th term exceeds the twice the third term by 1. Find the A.P.

Solution:

Given, 4th term of the A.P = thrice the first term

⟹ a4 = 3

first term Assuming first term to be ‘a’ and the common difference be ‘d’ we have,

a+ ( 4 -1 )d = 3

a ⟹ a + 3d = 3

a ⟹ a = 32d ... (1) and also it is given that, the 7th term exceeds the twice of the 3rd term by

1 ⟹ a7 + 1 = 2 × a3

⟹ a + (7 - 1)d + 1 = 2[a + (3 - 1)d]

⟹ a + 6d + 1 = 2a + 4d

⟹ a = 2d + 1   ... (2)

Putting the value of a = 32d from equation (1) in equation (2)

32d = 2d + 1

⟹ 32d - 2d = 1

⟹ 3d − 4d2 = 1

⟹ - d = 2

⟹ d = – 2 put d = -2 in a = 32d

⟹ a = 32(-2)

⟹ a = – 3

Now, we have a = – 3 and d = – 2, so the A.P. is -2, – 5, – 8, -11, …

Question: 7

Calculate the third term and the nth term of an A.P. whose 8th term and 13th term are 48 and 78 respectively.

Solution:

Given, a8 = 48 and a13 = 78 nth term of an A.P. is:

an = a + (n -1)d

so, a8 = a + (8 - 1)d = a + 7d   .... (1)

a13 = a + (13 - 1)d = a + 12d  .... (2)

Equating (1) and (2), we get.

⟹ a + 12d - (a + 7d) = 78 - 48

⟹ a + 12d - a -7d = 30

⟹ 5d = 30

⟹ d = 6

Putting the value of d = 6 in equation (1),

a + 7 × 6 = 48

⟹ a + 42 = 48

⟹ a = 4

Now, we have the first term (a) and the common difference (d) with us, So, nth term will be:

an = a + (n-1)d

= 4 + (n-1)6 = 4 + 6n - 6

an = 6n - 2 and the 3rd term will be

a3 = 6 × 3 - 2

an = 16

Question: 8

How many three digit numbers are divisible with 3?

Solution:

We know the first three digit number which is divisible by 3 is 102 and the last three digit number which is divisible by 3 is 999 So, here we have First term (a) = 102 Common Difference (c. d) = 3 last term or nth term (l) = 999

⟹ an = 999

⟹ a + (n - 1)c. d = 999

⟹ 102 + (n - 1)3 = 999

⟹ 102 + 3n - 3 = 999

⟹ 99 + 3n = 999

⟹ 3n = 900

⟹ n = 300

Therefore, there are 300 terms in the sequence.

Question: 9

An A.P. has 50 terms and the first term is 8 and the last term is 155. Find the 41stterm from the A.P.

Solution:

Given, First term (a) = 8 Number of terms (n) = 50 Last term (an) = 148

⟹ an = a + (n - 1)d

⟹ 155 = 8 + (50 - 1)d

⟹ 49 d = 147

⟹ d = 3 now, 41st term will be:

a + (41 - 1)d

⟹ 8 + 40 × 3

⟹ 128

Question: 10

The sum of 4th and 8th term of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

Solution:

Let’s assume first term be a and common difference be d

Given 4th term + 8th term = 24

⟹ a4 + a8 = 24

⟹ (a + (4 - 1)d) + (a + (8 - 1)d) = 24

⟹ a + 3d + a + 7d = 24

⟹ 2a + 10d = 24  .... (1)

And 6th term + 10th term = 34

⟹ a6 + a10 = 34

⟹ (a + 5d) + (a + 9d) = 34

⟹ 2a + 14d = 34  .... (2)

Subtracting equation (1) from (2), We get

⟹ (2a + 14d) - (2a + 10d) = 34 -24

⟹ 2a + 14d - 2a - 10d = 10

⟹ 4d = 10

⟹ d = 52 Put d = 52 in equation (1)

⟹ 2a + 10 × 52 = 24

⟹ 2a + 25 = 24

⟹ 2a = -1

⟹ a = −12

Therefore, we have a = −12 and d = 52

Question: 11

The first term of an A.P. is 7 and its 100th term is – 488, Find the 50th term of the same A.P.

Solution:

Given, First term

(a) = 7 100th term (a100) = -488

We know,

an = a + (n - 1)d

⟹ (a100) = a + (100 -1)d

⟹ 7 + 99d = - 488

⟹ 99d = - 495

⟹ d = - 5

Now, we have the common difference (d) = 5

We have to find out the 50th term of the A.P.

Then,

a50 = a + 49 d = 7 + 49 × (-5)

= 7 - 245 = – 238

So, the 50th term of the A.P. is – 238

Question: 12

Find a40 - a30 of the following A.P.

(i) 3, 5, 7, 9, . . .
(ii) 4, 9, 14, 19, . . .

Solution:

(i) Provided A.P. is 3, 5, 7, 9, . . . So, we have first term (a) = 3 and the common difference (d) is 5 - 3 = 2 we have to find

a40 - a30 = (a + 39d) - (a + 29d)

= a + 39d - a – 29d = 10d

= 10 × 2 = 20

(ii) Given A.P. is 4, 9, 14, 19, . . .

Common difference (d) = 9 - 4 = 5 we have to find

a40 - a30 = 10d = 10 × 5 = 50

Question: 13

Write the expression am - an for the A.P. a, a + d, a + 2d, . . .

Solution:

General Arithmetic Progression a, a + d, a + 2d, . . . am - an = (a + (m - 1)d) - ( a + ( n - 1 )d)

⟹ a + md - d - a - nd + d

⟹ md - kd

⟹ (m - n)d .... (1)

Hence find the common difference of the A.P. for which

(i) 11th term is 5 and 13th term is 79

Given, 11th term (a11) = 5 and 13th term (a­13) = 79 from equation (1), taking m = 11 and n = 13

⟹ am - an = (13 - 11)d

⟹ 79 - 5 = 2d

⟹ 74 = 2d

⟹ d = 37

(ii)  a10 - a5 = 200

Given, here we have the difference between the 10th term and 5th term.

Putting the value of m and n in equation (1) as 10 and 5, we have

⟹ a10 - a5 = (10 - 5 )d

⟹ 200 = 5 d

⟹ d = 40

(iii) 20th term is 10 more than the 18th term

Given,

a20 + 10 = a18

⟹ a20 - a18 = 10 from equation (1), we have

am - an = (m - n)d

⟹ a20 - a18 = (20 - 18) d

⟹ 10 = 2d

⟹ d = 5

Question: 14

Find n if the given value of x is the n term if the given A.P. (iii)  – 1, – 3, – 5, – 7, . . . : x = – 151

(iv) 25, 50, 70, 100, . . . : x = 1000

Solution:

(i) Given sequence is first term (a) = 1
Common difference (d) nth term an = a + (n-1) × d (ii) Given sequence is (iii) Given sequence is, -1 , -3, -5, -7, . . . :x = -151 first term (a) = -1

Common Difference (d) = -3 - (-1) = -3 + 1 = -2 nth term

an = a + (n - 1) × d

⟹ -151 = -1 + (n - 1) X - 2

⟹ -151 = -1 - 2n + 2

⟹ -151 = 1 - 2n

⟹ 2n = 152

⟹ n = 76

(iv) Given sequence is, 25, 50, 70, 100, . . . :x = 1000

First term (a) = 25

Common Difference (d) = 50 - 25 = 25 nth term an = a + (n - 1) × d we have

an = 1000

⟹ 1000 = 25 + ( n - 1 ) 25

⟹ 975 = ( n - 1)25

⟹ n - 1 = 39

⟹ n = 40

Question: 15

If an A.P. consists of n terms with the first term a and nth term 1. Show that the sum of the mth term from the beginning and the mth term from the end is (a + 1).

Solution:

First term of the sequence is a Last term (l) = a + (n - 1) d

Total no. of terms = n

Common Difference = d mth term from the beginning

am = a + ( n - 1 )d mth term from the end

= l + (n - 1)(-d)

⟹ a(n - m + 1) = l - (n - 1)d

⟹ am + a(n - m + 1)

= a + (n - 1)d + (l - ( n - 1)d)

= a + (n-1)d + l - (n - 1)d = a + l

Question: 16

Find the A.P. whose third term is 16 and seventh term exceeds its fifth term by 12.

Solution:

Given, a3 = 16

⟹ a + (3 - 1)d = 16

⟹ a + 2d = 16 … (i) and a7 - 12 = a5

⟹ a + 6d -12 = a + 4d

⟹ 2d = 12

⟹ d = 6 Put d = 6 in equation (1) a + 2 X 6 = 16

⟹ a + 12 = 16

⟹ a = 4.

So, the sequence is 4, 10, 16, . . .

Question: 17

The 7th term of an A.P is 32 and its 13th term is 62. Find the A.P.

Solution:

Given a7 = 32

⟹ a + (7 - 1)d = 32

⟹ a + 6d = 32  … (i) and a13 = 62

⟹ a + (13 - 1)d = 62

⟹ a + 12d = 62  … (ii) equation (ii) - (i), we have (a + 12d) - (a + 6d) = 62 - 32

⟹ 6d = 30

⟹ d = 5 Putting d = 5 in equation (i) a + 6 × 5 = 32

⟹ a = 32 - 30

⟹ a = 2

So, the obtained A.P. is 2, 7, 12, 17, . . .

Question: 18

Which term of the A.P. 3, 10, 17, . . . will be 84 more than its 13th term?

Solution:

Given A.P. is 3, 10 , 17, . . . First term (a) = 3

Common Difference (d) = 10 -3 = 7

Let nth term of the A.P. will be 84 more than its 13th term, then an = 84 + a13

⟹ a + (n - 1)d = a + (13 - 1)d + 84

⟹ (n - 1) × 7 = 12 × 7 + 84

⟹ n - 1 = 24

⟹ n = 25

Hence, 25th term, of the given A.P. is 84 more than the 13th term.

Question: 19

Two arithmetic progressions have the same common difference. The difference between their 100th terms is 100. What is the difference between their 1000th terms?

Solution:

Let the two A.P. be a1, a2 , a3 , . . .  and  b1, b2 , b3, . . . an = a1 + (n - 1)d and bn = a1 + (n - 1)d

Since common difference of two equation is same and given difference between 100th terms is 100

⟹ a100 - b100 = 100

⟹ a + (100 - 1)d - [b + (100 - 1)d] = 100

⟹ a + 99d -b - 99d = 100

⟹ a + b = 100  … (1)

Difference between 100th term is

⟹ a1000 - b­1000 = a + (1000 - 1)d - [b + (1000 - 1)d]

= a + 999d - b - 999d = a – b = 100  (from equation 1)

Therefore, Difference between 1000th terms of two A.P. is 100.

Question: 20

For what value of n, the nth terms of the Arithmetic Progression 63, 65, 67, . . . and , 3, 10, 17, . . . are equal?

Solution:

Given two A.P.s are: 63, 65, 67, . . . and 3, 10, 17, . . .

First term for first A.P. is (a) = 63

Common difference (d) is 65 - 63 = 2 nth term (an) = a + (n - 1)d = 63 + (n - 1)2

First term for second A.P.  is (a’) = 3

Common Difference (d’) = 10 - 3 = 7 nth term (a’n) = a’ + (n - 1)d =  3 + (n - 1) 7

Let nth  term of the two sequence be equal then, ⟹ 63 + (n - 1)2 = 3 + (n - 1)7

⟹ 60 = ( n - 1 ).7 - ( n - 1 ).2

⟹ 60 = 5(n - 1)

⟹ n - 1 = 12

⟹ n = 13

Hence, the 13th term of both the A.P.s are same.

Question: 21

How many multiple of 4 lie between 10 and 250?

Solution:

Multiple of 4 after 10 is 12 and multiple of 4 before 250 is 120/4, remainder is 2, so, 250 - 2 = 248

248 is the last multiple of 4 before 250 the sequence is 12,. . . . . . . . , 248 with first term (a) = 12

Last term (l) = 258

Common Difference (d) = 4 nth term (an) = a + (n - 1)d

Here nth term a n = 248

⟹ 248 = a + ( n - 1 )d

⟹ 12 + (n - 1)4 = 248

⟹ (n - 1)4 = 236

⟹ n - 1 = 59

⟹ n = 59 + 1

⟹ N = 60

Therefore, there are 60 terms between 10 and 250 which are multiples of 4

Question: 22

How many three digit numbers are divisible by 7?

Solution:

The three digit numbers are 100, . . . . . . , 999 105 us the first 3 digit number which is divisible by 7 and when we divide 999 with 7 remainder is 5, so, 999 - 5 = 994

994 Is the last three digit number which is divisible by 7.

The sequence here is 105, .. . . . . . . . , 994

First term (a) = 105 Last term (l) = 994

Common Difference (d) = 7

Let there are n numbers in the sequence then,

⟹ an = 994

⟹ a + (n - 1)d = 994

⟹ 105 + (n - 1)7 = 994

⟹ (n - 1) × 7 = 889

⟹ n - 1 = 127

⟹ n = 128

Therefore, there are 128 three digit numbers which are divisible by 7.

Question: 23

Which term of the A.P. 8, 14, 20, 26, . . . will be 72 more than its 41st term?

Solution:

Given sequence 8, 14, 20, 26, . . . Let its n term be 72 more than its 41st term

⟹ an = a41  + 72   … (1)

For the given sequence, first term (a) = 8,

Common Difference (d) = 14 - 8 = 6 from equation (1), we have

an = a41 + 72

⟹ a + (n - 1)d = a + (41 - 1)d + 72

⟹ 8 + (n - 1)6 = 8 + 40 × 6 + 72

⟹ (n - 1)6 = 312

⟹ n - 1 = 52

⟹ n = 53

Therefore, 53rd term is 72 more than its 41st term.

Question: 24

Find the term of the Arithmetic Progression 9, 12, 15, 18, . . . which is 39 more than its 36th term.

Solution:

Given A.P. is 9, 12, 15, 18 , . . .

Here we have, First term (a) = 9

Common Difference (d) = 12 - 9 = 3

Let its nth term is 39 more than its 36th term So,

an­  = 39 + a36

⟹ a + (n - 1)d = 39 + a + (36 - 1)d

⟹ (n - 1)3 = 39 + 35 × 3

⟹ (n - 1)3 = (13 + 35)3

⟹ n - 1 = 48

⟹ n = 49

Therefore, 49th term of the A.P. 39 more than its 36th term.

Question: 25

Find the 8th term from the end of the A.P. 7, 10, 13, . . . , 184.

Solution:

Given A.P. is 7, 10, 13, . . . , 184

First term (a) = 7

Common Difference (d) = 10 - 7 = 3 last term (l) = 184 nth term from end = l - ( n - 1 )d

8th term from end = 184 - ( 8 - 1 )3 =

184 - 7 X 3 =

184 - 21 = 183

Therefore, 8th term from the end is 183

Question: 26

Find the 10th term from the end of the A.P. 8, 10, 12, . . . , 126

Solution:

Given A.P. is 8, 10, 12, . . . , 126

First term (a) = 8

Common Difference (d) = 10 - 8 = 2

Last term (l) = 126 nth term from end is: l - (n - 1)d

So, 10th term from end is: l - (10 - 1)d

= 126 - 9 × 2 = 126 - 18 = 108

Therefore, 109 is the 10th term from the last in the A.P. 8, 10, 12, . . 126.

Question: 27

The sum of 4th and 8th term of an A.P. is 24 and the sum of 6th and 10th term is 44. Find the Arithmetic Progression.

Solution:

Given a4 + a8 = 24

⟹ a + (4 - 1)d + a + (8 - 1)d = 24

⟹ 2a + 3d + 7d = 24

⟹ 2a + 10d = 24  … (1) and a6 + a10 = 44

⟹ a + (6 - 1)d + a + (10 - 1)d = 44

⟹ 2a + 5d + 9 d = 44

⟹ 2a + 14d = 44   … (2) equation (2) - equation (1), we get

2a + 14d - (2a + 10d) = 44 - 24

⟹ 4d = 20

⟹ d = 5 Put d = 5 in equation (1), we get

2a + 10 × 5 = 24

⟹ 2a = 24 - 50

⟹ 2a = - 26

⟹ a = -13 The A.P is -13, - 7, – 2, . . .

Question: 28

Which term of the A.P. is 3, 15, 27, 39, . . . will be 120 more than its 21st term?

Solution:

Given A.P. is 3, 15, 27, 39, . . . First term (a) = 3

Common Difference (d) = 15 - 3 = 12

Let nth term is 120 more than 21st term

⟹ an = 120 + a21

⟹ a + (n - 1)d = 120 + a + ( 21 - 1 )d

⟹ (n - 1)d = 120 + 20d

⟹ (n - 1)12 = 120 + 20 × 12

⟹ n - 1 = 10 + 20

⟹ n = 31

Therefore, 31st term of the A.P. is 120 more than the 21st term.

Question: 29

The 17th term of an A.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43. Find the nth term.

Solution:

Given 17th term of an A.P is 5 more than twice its 8th term

⟹ a17 = 5 + 2a8

⟹ a + (17 - 1)d = 5 + 2 [a + (8 - 1)d]

⟹ a + 16d = 5 + 2a + 14d

⟹ a + 5 = 2d  … (1) and 11th term of the A > P. is 43 a11 = 43

⟹ a + (11 - 1)d = 43

⟹ a + 10d = 43

⟹ a + 5 × 2d = 43 from equation (1)

⟹ a + 5 × (a + 5) = 43

⟹ a + 5a + 25 = 43

⟹ 6a = 18

⟹ a = 3

Putting the value of a = 3, in equation (1), we get 3 + 5 = 2d

⟹ 2d = 8

⟹ d = 4

We have to find the nth term (an) = a + (n - 1)d

= 3 + (n - 1)4 = 3 + 4n - 4 = 4n - 1

Therefore, nth term is 4n - 1

Question: 30

Find the number of all three digit natural number which are divisible by 9.

Solution:

First three-digit number that is divisible by 9 is 108. Next number is 108 + 9 = 117.

And the last three-digit number that is divisible by 9 is 999.

Thus, the progression will be 108, 117, .... , 999.

All are three digit numbers which are divisible by 9, and thus forms an A.P. having first term

(a): 108 last term (l) = 999 and the common difference (d) as 9 We know that, nth term

(an) = a + (n - 1)d

According to the question,

999 = 108 + (n - 1)9

⟹ 999 = 108 + 9n - 9

⟹ 999 = 99 + 9n

⟹ 999 = 9n

⟹ 999 - 99

⟹ 9n = 900

⟹ n = 100

Therefore, There are 100 three digit terms which are divisible by 9.

Question: 31

The 19th term of an A.P. is equal to three times its 6th term. if its 9th term is 19, find the A.P

Solution:

Let a be the first term and d be the common difference.

We know that, nth term

= an= a + (n - 1)d

According to the question, a19 = 3a6

⟹ a + (19 - 1)d = 3(a + (6 - 1)d)

⟹ a + 18d = 3a + 15d

⟹ 18d- 15d = 3a - a

⟹ 3d = 2a

⟹ a = 32d  .... (1) Also, a9 = 19

⟹ a+(9 - 1)d= 19

⟹ a + 8d = 19 ... (2)

On substituting the values of (1) in (2), we get

⟹ 32d + 8d = 19

⟹ 3d + 16d = 19 × 2

⟹ 19d = 38

⟹ d = 2 Now, a = 32 × 2  [From (1)] a = 3 Therefore, The A.P. is : 3, 5, 7, 9, . . .

Question: 32

The 9th term of an A.P. is equal to 6 times its second term. If its 5th term is 22, find the A.P.

Solution:

Let a be the first term and d be the common difference.

We know that, nth term (an) = a + (n - 1)d

According to the question, a9 = 6a2

⟹ a + (9 - 1)d = 6(a + (2 - 1)d)

⟹ a + 8d = 6a + 6d

⟹ 8d - 6d = 6a - a

⟹ 2d = 5a

⟹ a = 25  .... (1) Also, a5 = 22

⟹ a + (6 - 1)d = 22

⟹ a + 4d = 22  ... (2)

On substituting the values of (1) in (2), we get

2/5 d + 4d = 22

⟹ 2d + 20d = 22 × 5

⟹ 22d = 110

⟹ d = 5 Now,  a = 25 × 5  [From (1)] ⟹ a = 2

Thus, the A.P. is : 2, 7, 12, 17, . . .

Question: 33

The 24th term of an A.P. is twice its 10th term. Show that its 72nd term is 4 times its 15th term.

Solution:

Let a be the first term and d be the common difference.

We know that, nth term

(an) = a + (n - 1)d

According to the question, a24 = 2

a10 ⟹ a + (24 - 1)d = 2(a + (10 - 1)d)

⟹ a + 23d = 2a + 18d

⟹ 23d - 18d = 2a - a

⟹ 5d = a

⟹ a = 5d  .... (1)

Also, a­72 = a + (72 - 1)

d = 5d + 71d  [From (1)] = 6d ….(2) and a15 = a + (15 - 1) d = 5d + 14d [From (1)] = 19d …. (3)

On comparing (2) and (3), we get

⟹ 76d = 4 × 19 d

⟹ a72 = 4 × a15

Thus, 72nd term of the given A.P. is 4 times its 15th term.

Question: 34

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

Solution:

Since, the number is divisible by both 2 and 5, means it must be divisible by 10.

In the given numbers, first number that is divisible by 10 is 110.

Next number is 110 + 10 = 120.

The last number that is divisible by 10 is 990.

Thus, the progression will be 110, 120, ..., 990.

All the terms are divisible by 10, and thus forms an A.P. having first term as 110 and the common difference as 10.

We know that, nth term = an= a + (n - 1)d

According to the question, 990 = 110 + ( n - 1 )10

⟹ 990 = 110 + 10n - 10

⟹ 10n = 990 - 100

⟹ n = 89

Thus, the number of natural numbers between 101 and 999 which are divisible by both 2 and 5 is 89.

Question: 35

If the7th term of an A.P. is 1/9 and its 9th term is 1/7, find the 63rd term.

Solution:

Let a be the first term and d be the common difference.

We know that, nth term = an= a + (n - 1)d

According to the question, a7 = 1/9

⟹ a+(7 - 1)d = 1/9

⟹ a+ 6d = 1/9   ....(1) Also, a9 = 1/7

⟹ a + (9 - 1)d = 1/7

⟹  a + 8d = 1/7  ....(2)

On Subtracting (1) from (2), we get

⟹ 8d - 6d = 1/7 - 1/9

⟹2d = (9 - 7)/63

⟹ 2d = 2/63

⟹ d = 1/63

Put value of d = 1/63 in equation (1), we get

⟹ a + 6 × 1/63 = 1/9

⟹ a = 1/9 - 6/63

⟹a = (7 - 6)/63

⟹ a = 1/63

Therefore, a63 = a + (63 - 1)d = 1/63 + 62/63 = 63/63 = 1

Thus, 63rd term of the given A.P. is 1.

Question: 36

The sum of 5th and 9th terms of an A.P. is 30. If its 25th term is three times its 8th term, Find the A.P.

Solution:

Let a be the first term and d be the common difference.

We know that, nth term (an) = a + (n - 1)d

According to the question, a5 + a9 = 30

⟹ a + (5 - 1)d + a + (9 - 1)d = 30

⟹ a + 4d + a + 8d = 30

⟹ 2a + 12d = 30

⟹ a + 6d = 15  .... (1) Also, a25 = 3(a8)

⟹ a + (25 - 1)d = 3[a + (8 - 1)d]

⟹ a + 24d = 3a + 21d

⟹ 3a - a = 24d - 21d

⟹ 2a = 3d

⟹ a = 32d   ….(2)

Substituting the value of (2) in (1), we get 32d + 6d = 15

⟹ 32d + 6d = 15

⟹ 3d + 12d = 15 × 2

⟹ 15d = 30

⟹ d = 2 now, a = 32d × 2  [From (1)] ⟹ a = 3 Therefore, the A.P. is 3, 5, 7, 9, . . .

Question: 37

Find whether 0 (zero) is a term of the A.P. 40, 37, 34, 31, . . .

Solution:

Let a be the first term and d be the common difference.

We know that, nth term = an = a + (n - 1)d

It is given that a = 40, d = -3 and an= 0

According to the question, ⟹ 0 = 40 + (n - 1)(-3)

⟹ 0 = 40 - 3n + 3

⟹ 3n = 43

⟹ n = 433 .... (1)

Here, n is the number of terms, so must be an integer.

Thus, there is no term where 0 (zero) is a term of the A.P. 40, 37, 34, 31,. . .

Question: 38

Find the middle term of the A.P. 213, 205, 197, . . . 37.

Solution:

Let a be the first term and d be the common difference. We know that, nth term (an) = a + (n - 1)d

It is given that a = 213, d = – 8 and an = 37

According to the question, ⟹ 37 = 213 + (n - 1)(-8)

⟹ 37 = 213 - 8n + 8

⟹ 8n = 221 - 37

⟹ an = 184

⟹ n = 23  .... (1)

Therefore, total number of terms is 23. Since, there is odd number of terms.

So, Middle term will be 23 + 12th term, i.e., the 12th term.

a12 = 213 + (12 - 1)(-8)

a12 = 213 - 88 = 125

Thus, the middle term of the A.P.  213, 205, 197, . . . ,  37 is 125.

Question: 39

If the 5th term of the A.P. is 31 and 25th term is 140 more than the 5th term, find the A.P.

Solution:

Let a be the first term and d be the common difference.

We know that, nth term (an) = a + (n - 1)d

According to question, a6 = 31

⟹ a + (5 - 1) = 31

⟹ a + 4d = 31

⟹ a = 31 - 4d  .... (1)Also, a25  = 140 + a5

⟹ a + (25 - 1) = 140 + 31

⟹ a + 24d = 171 .... (3)

On substituting the values of (1) in (2), we get

31 - 4d + 24d = 171

⟹ 20d = 171 - 31

⟹ 20d = 140

⟹ d = 7

⟹ a = 31 - 4 × 7 [From (1)]

⟹ a = 3 Thus, the A.P. obtained is 3, 10, 17, 24, . .
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