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If the sum of the first $p$ terms of an A.P. is $q$ and the sum of first $q$ terms is $p$ , then find the sum of first $(p + q)$ terms from the following options is:

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p+q

–(p+q)

answer is D.

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Detailed Solution

Given that, sum of

p

terms A.P is

q

and the sum of first

q

terms are

p

.
The sum of the first terms is given b,

S u m = n 2 2 a + n − 1 d

The sum of

p

terms of an A.P is

q

S u m = n 2 2 a + n − 1 d ⇒ q = p 2 2 a + p − 1 d ⇒ 2 q = 2 a p + p p − 1 d ...... (1)

The sum of

q

terms of an A.P is

p

S u m = n 2 2 a + n − 1 d ⇒ p = q 2 2 a + q − 1 d ⇒ 2 p = 2 a q + q q − 1 d ...... (2)

Equating equation (1) and (2) and find the value of first term

a

and common difference

d

.
Multiplying

q

in (1) and

p

in (2) and then subtracting equation (1) and (2)

⇒ 2 q 2 = 2 a p q + p q p − 1 d

⇒ 2 p 2 = 2 a p q + p q q − 1 d

We get,

⇒ 2 q 2 − 2 p 2 = d p 2 q − p q − p q 2 + p q ⇒ d = 2 q 2 − 2 p 2 p 2 q − p q 2 ⇒ d = 2 q − p q + p p q p − q ⇒ d = − 2 p + q p q ...... (3)

From equation (1),

⇒ 2 q = 2 a p + p p − 1 d ⇒ 2 a p = 2 q + p p − 1 2 p + q p q ⇒ a p = q + p + q p − 1 q ⇒ a p = q 2 + p 2 − p + p q − q q

So, the value of

a = q 2 + p 2 − p + p q − q p q

……(4)
Sum of

p + q

term is,

⇒ Sum = p + q 2 2 a + p + q − 1 d

Put the value of

a, d

from equation (3) and (4) Question Image

The sum of

p + q th term is − p + q

.
Hence, option 4) is correct.

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If the sum of the first p terms of an A.P. is q and the sum of first q terms is p , then find the sum of first (p+q) terms from the following options is:

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If the sum of the first $p$ terms of an A.P. is $q$ and the sum of first $q$ terms is $p$ , then find the sum of first $(p + q)$ terms from the following options is: