### Question 11. Find the sum of all integers between 100 and 550 which are divisible by 9.

**Solution:**

According to question

A.P = 108, 117…549

From the given A.P. we geta(first term) = 108, d(common difference) = 9, a

_{n}(nth term) = 549

Find the value of n using the given formulaa

_{n}= a + (n – 1)d549 = 108 + (n – 1)(9)

441 = 9n – 9

450 = 9n

n = 50

Now, we find the sum of the given A.P. using the following formulaS = n/2 [2a + (n -1)d]

S = 50/2 [2(108) + (50 – 1)(9)]

= 25 [216 + (49)(9)]

= 25 [216 + 441]

= 25 [657]

= 16425

Hence, the the sum of all integers between 100 and 550 which are divisible by 9 = 16425

### Question 12. Find the sum of the series:

### 3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + … to 3n terms.

**Solution:**

A.P = 3 +5 +7 + 9 + … to 3n

From the given A.P. we geta = 3, d = 2, n = 3n

Now, we find the sum of the given A.P. using the following formulaS = n/2 [2a + (n – 1)d]

= 3n/2 [2(3) + (3n – 1)(2)]

= 3n [3 + (3n – 1)]

= 3n [3n + 2]

Hence, the sum of the given A.P = 3n [3n + 2]

### Question 13. Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.

**Solution:**

According to question

A.P = 103, 119,…,791

From the given A.P. we geta = 103, l = 791

Find the value of n using the given formulaa

_{n}= a + (n – 1)d791 = 103 + (n – 1)16

n = 44

Now, we find the sum of the given A.P. using the following formulaS = n/2 [a + l]

S = 44/2 [103 + 791]

= 22 [894]

= 19668

Hence, the sum of all those integers between 100 and 800

each of which on division by 16 leaves the remainder 7 = 19668

### Question 14. Solve:

### (i) 25 + 22 + 19 + 16 + … + x = 115

### (ii) 1 +4+7+ 10 + … + x = 590

**Solution:**

(i)A.p= 25 + 22 + 19 + 16 +…+x = 115

From the given A.P. we geta = 25, d = -3, S = 115

Using the formula

S = n/2[2a + (n – 1)d]

⇒ 115 = n/2 [2 x 25+ (n – 1)(-3)]

⇒ 115 x 2 = n[50 – 3n + 3]

⇒ 230 = n(53 – 3n)

⇒ 230 = 53n – 3n

^{2}⇒ 3n

^{2}– 53n + 230 = 0Using quadratic formula:

Now, Put the value of a = 3, b = – 53 and c = 230, we get

n = 46/6,10

⇒ n = 10 as n ≠ 46/6

So, a

_{n}= x = a + (n – 1)d⇒ x = 25 + (10 – 1)(-3)

⇒ x = 25 – 27 = -2

Hence, the value of x = -2

(ii)A.P =1 + 4 + 7 + 10 +….+ x = 590

From the given A.P. we geta = 1, d = 3

Using the formula

S = n/2 [2a + (n – 1)d]

⇒ 590 = n/2[2 x 1 + (n – 1)(3)]

⇒ 590 x 2 = n[2 + 3n – 3]

⇒ 1180 = n(3n – 1)

⇒ 1180 = 3n

^{2}– n⇒ 3n

^{2}– n -1180 = 0Using quadratic formula:

Now, Put the value of a = 3, b = – 1 and c = -1180, we get

n = -118/6, 20

⇒ n = 20, as n ≠ -118/6

a

_{n}= x = a + (n – 1)d⇒ x = 1 + (20 – 1)(3)

⇒ x = 1 + 60 – 3 = 58

Hence, the value of x = 58

### Question 15. Find the r^{th} term of an A.P., the sum of whose first n terms is 3n^{2} + 2n.

**Solution:**

According to the question

As, S

_{n}= 3n^{2}+ 2nSo, a = S

_{1}= 3 x 1^{2}+ 2 x 1 = 3 + 2 = 5 andS

_{2}= 3 x 2^{2}+2 x 2 = 12+4 = 16⇒ a + a2 = 16

⇒ a + a + d = 16

⇒ 2a + d = 16

⇒ 2 x 5 + d = 16

⇒ d = 16 – 10

⇒ d = 6

Now,

a

_{r }= a + (r – 1)d= 5+ (r -1) x 6

= 5 + 6r – 6

Hence, the a

_{r }= 6r – 1

### Question 16. How many terms are there in the A.P. whose first and fifth terms are -14 and 2 respectively and the sum of the terms is 40?

**Solution:**

According to question we have

a = -14 and S

_{n}= 40 ………………………. (i)a

_{5}= 2By using formula

⇒ a + (5 – 1)d = 2

⇒ -14 + 4d = 2

⇒ 4d = 16

⇒ d = 4 …………………………..(ii)

By using formula

S

_{n}= n/2 [2a + (n – 1)d]⇒ 40 = n/2 [2(-14) + (n – 1) x 4] ( From eq(i) and (ii))

⇒ 80 = n[-28 + 4n – 4]

⇒ 80 = 4n

^{2}– 32n⇒ n

^{2}– 8n – 20 = 0⇒ (n -10)(n + 2) = 0

⇒ n = 10, -2

But n cannot be negative.

Hence, the total number of terms in the A.P = 10

### Question 17. The sum of the first 7 terms of an A.P. is 10 and that of the next 7 terms is 17. Find the progression.

**Solution: **

According to question we have

S

_{7}= 10By using formula

S

_{n}= n/2 [2a + (n – 1)d]⇒ 7/2 [2a + (7 – 1)d] = 10

⇒ 7/2 [2a + 6d] = 10

⇒ a + 3d = 10/7 ………………………………… (i)

Also, the sum of the next seven terms = S

_{14}– S_{7}= 17⇒ 14/2 [2a + (14 – 1)d] – 7/2 [2a + (7 – 1)d] = 17

⇒ 7[2a + 13d] – 7/2[2a + 6d] = 17

14a + 91d – 7a – 21d = 17

7a + 70d = 17

a + 10d = 17/7 ………………………………….. (ii)

From eq(i) and (ii), we get:

10/7 – 3d = 17/7 – 10d

⇒ 7d = 1

⇒ d = 1/7

On putting the value in eq(i), we get:

a + 3d = 10/7

⇒ a+ 3/7 = 10/7

⇒ a = 1, d = 1/7

Hence, the progression = 1, 8/7, 9/7, 10/7 …

### Question 18. The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference, and the sum of first 20 terms.

**Solution:**

According to question we have

a

_{3}= 7, a_{7}– 3a_{3}= 2By using formula

a

_{n}= a + (n – 1)d⇒ a + (3 – 1)d = 7

⇒ a + 2d = 7 …………………………………… (i)

Also,

a

_{7}– 3a_{3}= 2⇒ a

_{7}– 21 = 2(Given)⇒ a + (7 – 1)d = 23

⇒ a + 6d = 23 ………………………………………. (ii)

From eq(i) and (ii), we get

4d = 16

⇒ d = 4

On putting the value in eq(i), we get

a + 2(4) = 7

a = -1

By using formula

S

_{n}= n/2 [2a + (n – 1)d]S

_{20}= 20/2 [2(-1) + (20-1)(4)]⇒ S

_{20 }= 10[-2 + 76]⇒ S

_{20}= 10[74] = 740Hence,

a = -1, d = 4, S

_{20}= 740

### Question 19. The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

**Solution:**

According to question we have

a = 2,l = 50, S

_{n}= 442Now, S

_{n}= 442⇒ n/2[a + l] = 442

⇒ n/2 [2 + 50] = 442

⇒ n = 17

a

_{17}= 50By using formula, we get

⇒ a + (17 – 1) d = 50

⇒ 2 + 16d = 50

⇒ d = 3

### Question 20. The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by , find the number of terms and the series.

**Solution:**

According to question, we have

a

_{1}+ a_{3}+… +a_{2n-1}= 24 ……………..(1)a

_{2}+ a_{4}+… +a_{2n}= 30 ……………..(2)Now, subtracting eq(1) from (2), we get:

(d + d+…+ upto n terms) = 6

⇒ nd = 6 ………….(3)

Given :

a

_{2n}= a_{1}+ 21/2⇒ a

_{2n }– a_{1}= 21/2⇒ a + (2n – 1)d – a = 21/2 [a

_{2n}= a + (2n – 1)d, a_{1 }= a]⇒ 2nd – d = 21/2

⇒ 2 x 6 – d = 21/2 (From eq(3))

⇒ d = 3/2

On putting the value in eq(3), we get

n = 4

⇒ 2n = 8

Thus, there are 8 terms in the progression.

To find the value of the first term:

a

_{2}+ a_{4}+…+a_{2n }= 30⇒ (a + d) + (a + 3d)+… +[a + (2n – 1)d] = 30

⇒ n/2 [(a + d) + a + (2n – 1)d] = 30

On putting n = 4 and d = 3/2, we get

a = 3/2

Hence, the series will be 1, 1/2, 3, …

### Question 21. If S_{n} = n^{2}p and S_{m} = m^{2} p, m ≠ n, in an A.P., prove that S_{p} = p^{3}.

**Solution:**

S_{n}= n^{2}p⇒ n/2 [2a + (n – 1)d] = n

^{2}p⇒ 2np = 2a + (n – 1)d ………………. (i)

S

_{m}= m^{2}p⇒ n/2 [2a + (m – 1)d] = m

^{2}p⇒ 2mp = 2a + (m – 1)d ……………….. (ii)

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

2p(n – m) = (n – m)d

2p = d ………………..(iii)

On substituting the value in eq(i), we get

nd = 2a + (n – 1)d

⇒ nd – nd + d = 2a

⇒ a = d/2 = p [from eq(iii)] ……………….(iv)

S

_{p }= p/2[2a + (p – 1)d]⇒ S

_{p}= p/2[2p + (p – 1)2p]⇒ S

_{p}= p/2[2p + 2p^{2}-2p]⇒ S

_{p}= p/2[2p^{2}]⇒ S

_{p}= p^{3}Hence Proved

### Question 22. If 12th term of an A.P. is -13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

**Solution:**

Let us considered first term = a and common difference = d

Given that a

_{12}= -13Using the formula

⇒ a + (12 – 1)d = -13

⇒ a + 11d = -13 ………………. (i)

Also, S

_{4}= 24 (given)Using the formula

⇒ 4/2 [2a + (4 – 1)d] = 24

⇒ 2(2a + 3d) = 24

⇒ 2a + 3d = 12 ………………. (ii)

From eq(i) and (ii), we get

19d = -38

d = -2

Now put the value of d in eq(i), we get

a + 11(-2) = -13

⇒ a = 9

S

_{10 }= 10/2 [(2)(9) + (10 – 1)(-2)]⇒ S

_{10 }= 5[18 – 18] = 0Hence, the sum of first ten terms = 0