Class 11 RD Sharma Solutions - Chapter 19 Arithmetic Progressions- Exercise 19.4 | Set 3

Question 23. If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of the first 20 terms?  

Solution:

According to question we have

a₅ = 30 

Using the formula

⇒ a + (5 - 1)d = 30 

⇒ a + 4d = 30        ................ (i) 

Also, a_n = 65 

Using the formula

⇒ a + (12 - 1)d = 65 

⇒ a + 11d = 65      ...............(ii) 

On solving eq(i) and (ii), we get 

7d = 35 

⇒ d = 5 

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

a + 4 x 5 = 30 

⇒ a = 10 

Now we find the sum of first 20 terms

S₂₀ = 20/2 [2 x 10 + (20 - 1) x 5] 

⇒ S₂₀ = 10 [20 + 95]

⇒ S₂₀ = 1150

Question 24. Find the sum of n terms of the A.P. whose kth terms is 5k + 1.  

Solution:

According to question we have

a_k = 5k + 1 

For k = 1, a₁ = 5 x 1 + 1 = 6 

For k = 2, a₂ = 5 x 2 + 1 = 11 

For k = n, a_n = 5n + 1 

Now we find the sum of n terms

Sn = n/2 [a + an] 

⇒ Sn = n/2[6 + 5n + 1] = n/2 (5n + 7)  

Question 25. Find the sum of all two-digit numbers which when divided by 4, yields 1 as remainder. 

Solution:

According to question we have

A.P = 13, 17....97

From the A.P we have

a = 13, d = 4, a_n = 97 

Now using the formula

⇒ a_n = a (n - 1)d 

⇒ 97 = 13 + (n - 1)4 

⇒ 84 = 4n - 4 

⇒ 88 = 4n 

⇒ 22 = n          ............... (1) 

Now we find the sum of all two-digit numbers using the given formula

S_n = n/2[2a + (n - 1)d] 

S₂₂ = 22/2 [2 x 13 + (22 - 1) x 4]      ( From eq(1)) 

⇒ S₂₂ = 11[26 + 84]

⇒ S₂₂ = 11[110] = 1210

Question 26. If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term. 

Solution:

According to question 

A.P. = 25, 22, 19 

From the A.P we have

a = 25, d = 22 - 25 = -3 

S_n = 116 

Now using the formula

⇒ n/2 [2a + (n - 1)d] = 116 

⇒ n [2 x 25 + (n - 1)(-3)] = 232 

⇒ 50n -3n² + 3n = 232 

⇒ 3n² - 53n + 232 = 0 

⇒ 3n² - 29n - 24n + 232 = 0 

⇒ n(3n - 29) - 8(3n - 29) = 0 

⇒(3n - 29)(n - 8) = 0 

⇒ n = 29/ 3 or 8 

Since n cannot be a fraction, n = 8. 

Now we find the last term using the following formula

a_n = a + (n - 1)d

⇒ a₈ = 25 + (8 - 1)(-3)

⇒ a₈ = 4

Question 27. Find the sum of odd integers from 1 to 2001.  

Solution:

According to question 

A.P = 1, 3, 5 ... 2001

From the A.P we have

a = 1 and d = 2 

a_n = 2001 

Now using the formula

⇒ 1 + (n - 1)2 = 2001 

⇒ 2n - 2 = 2000 

⇒ 2n = 2002 

⇒ n = 1001  

Now we find the sum of odd integers from 1 to 2001

Also, S₁₀₀₁= 1001/2 [2 x 1 + (1001 - 1)2]

⇒ S₁₀₀₁ = 1001/2 [2 + 2000]

⇒ S₁₀₀₁ = 1001 x 1001 = 1002001 

Question 28. How many terms of the A.P. -6, -11/2, -5, ... are needed to give the sum -25? 

Solution:

According to question 

A.P = -6, -11/2, -5, ... 

From the A.P we have

a = - 6 and d =-11/2 - (-6) = 1/2

S_n = -25  

Now using the formula

 ⇒ -25 = n/2 [2 x (-6) + (n -1)(1/2)]

⇒ -25 = n/2 [ -12 + n/2 - 1/2]

⇒ -50 = n [ n/2 - 25/2]

⇒ -100 = n [ n - 25]

⇒ n² - 25n + 100 = 0

⇒ (n - 20)(n - 5) = 0

⇒ n = 20 or n = 5

Question 29. In an A.P. the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is -112.  

Solution:

According to question we have

a = 2, S₅ = 1/4(S₁₀ - S₅)  

S₅ =  5/2 [2 x 2 + (5 - 1)d]  

⇒ S₅ = 5 [2 + 2d]    ....................(i) 

Also, S₁₀ = 10/2 [2 x 2 + (10 - 1)d] 

⇒ S₁₀ = 5[4 + 9d]   ....................(ii) 

Since S₅ = 1/4 (S₁₀ - S₅) 

So, from eq(i) and (ii), we have: 

⇒ 5[2 + 2d] = 1/4 [5(4 + 9d) - 5(2 + 2d)] 

⇒ 8 + 8d = 4 + 9d - 2 - 2d  

⇒ d = -6

Now we find the 20th term

a₂₀ = a + (20 - 1)d 

⇒ a₂₀ = a + 19d

⇒ a₂₀ = 2 + 19(-6) = -112

Hence proved

Question 30. If S₁ be the sum of (2n + 1) terms of an A.P. and S₂ be the sum of its odd terms, the prove that: S₁ : S₂ = (2n + 1) : (n + 1) 

Solution:

Let us assume A. P. be a, a + d, a + 2d... 

So, S₁ = (2n + 1)/2 [2a + (2n + 1 - 1)d] 

Now using the formula, we get

⇒ S₁ = (2n + 1)/2 [2a + (2n)d] 

⇒ S₁ = (2n + 1)(a + nd)          .....................(i) 

Now using the formula, we get

S₂ = (n + 1)/2 [2a + (n + 1 - 1) x 2d] 

⇒ S₂ = (n +1)/2 [2a + 2nd] 

⇒ S₂ = (n + 1)[a + nd]           .......................(ii)

 From eq(i) and (ii), we get : 

S₁ / S₂ = (2n + 1) / (n + 1) 

Hence, proved 

Question 31. Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms. 

Solution:

Given that S_n = 3n²

So, for n = 1, S₁ = 3 x 1² = 3 

For n = 2, S₂ = 3 x 2² = 12 

For n = 3, S₃= 3 x 3² = 27 and so on  

So, S₁ = a₁ = 3 

a₂ = S₂ - S₁ = 12 - 3 = 9 

a3 = S₃ - S₂ = 27 - 12 = 15 and so on  

Hence, the A.P. = 3, 9, 15... 

Question 32. If the sum of n terms of an A.P. is nP + 1/2 - n (n - 1) Q, where P and Q are constants, find the common difference.

Solution:

According to question we have

S_n = nP + 1/2 n(n - 1)Q 

For n = 1, S₁ = P + 0 = P 

For n = 2, S₂ = 2P + Q 

Also, a₁ = S₁ = P,

a₂ = S₂  -  S₁ = 2P + Q - P = P + Q 

Hence, the common difference d = a₂ - a₁ = P + Q - P = Q  

Question 33. The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.

Solution:

Let us considered we have two A.P's. So, a₁ and a₂ are the first terms and d₁ and d₂ is common difference of the A.P's

According to question we have

(5n + 4) / (9n + 6) = (Sum of n terms in the first A.P.) / (Sum of n terms in the second A.P.) 

⇒ (5n + 4) / (9n + 6) = (2a₁+ [(n — 1)d₁]) / (2a₂ + [(n — 1)d₂]  .........(1)

Now put n = 2 x 18 - 1 = 35 in eq(1), we get 

(5 x 35 + 4) / (9 x 35 + 6) = (2a₁ + 34d₁) / (2a₂ + 34d₂)

179 /321 = (a₁ + 17d₁) / (a₂ + 17d₂) = (18th term of the first A.P.) / (18th term of the second A.P.)

Hence, the ratio of 18th terms = (a₁ + 17d₁) / (a₂ + 17d₂) = 179 /321

Question 34. The sums of n terms of two arithmetic progressions are in the ratio 7n + 2 : n + 4. Find the ratio of their 5th terms.

Solution:

Let us considered we have two A.P's. So, a₁ and a₂ are the first terms and S₁ and S₂ are the sum of the first n terms. 

According to question we have

S₁ = n/2 [2a₁ + (n - 1)d₁] 

And, S₂ = n/2 [2a₂ + (n -1)d₂] 

Given:  

S₁ / S₂ = (n/2 [2a₁ + (n - 1)d₁])/(n/2 [2a₂ + (n - 1)d₂]) = (7n + 2) / (n + 4)

Now we find the ratio of their 5th terms

[2a₁ + (9 - 1)d₁] / [2a₂ + (9 - 1)d₂] = (7(9) + 2) / (9 + 4)

[2a₁ + (8)d₁] / [2a₂ + (8)d₂] = 65/13

[a₁ + (4)d₁] / [a₂ + (4)d₂] = 5/1 = 5 : 1

Hence, the ratio of 5th terms = 5 : 1