Question

If the sum of first p terms of an ap is q and the sum of first q terms is p

Answer 1

Answer:

If the sum of first p terms of an AP is q and the sum of its first q terms is p then show that the sum of its first (p+q) terms is -(p+q).

Step-by-step explanation:Answer.

Let a be the first term and d is common difference of the A.P then sum of n terms in A.P

is Sn = (n/2)[ 2a + (n - 1) d]

Given that Sp = q and Sq = p.

Sp = (p/2)[ 2a + (p - 1) d] = q ⇒  [ 2a + (p - 1) d] = 2q / p --------(1)

Sq = (q/2)[ 2a + (q - 1) d] = p ⇒  [ 2a + (q - 1) d] = 2p / q --------(2)

Subtract (1) from (2) we get

(q - p)d = (2p / q) – (2q / p)

(q - p)d = (2p2– 2q2) / pq

d = -2(q +p) / pq  -----------(3)

Sum of first ( p + q ) terms

Sp +q = (p+ q) / 2 [ 2a + ( p + q -1) d]

Sp +q = (p+ q) / 2 [ 2a + ( p -1)d + qd]

Sp +q = (p +q) / 2 [ (2q /p)  + q(-2(q +p) / pq )]            [ from (1)  and (3)]

Sp +q = (p + q) / 2 [ (2q -2q-2p) / p )]

Sp +q = (p + q) / 2 [-2p) / p )]

Sp +q = -  (p + q)

hope it helps u

Answer 2

Answer:

Step-by-step explanation:

Hope this helps you!