Q13/059) If the sum of first p terms of an A.P. is q and the sum of first q terms is p, show that the sum of its first (p + q) terms is – (p + q).

let 1st term be a and common difference be d

so pth term = a + (p-1) d and sum of 1st p terms = ( 2a + (p-1)d)* (p/2) = q

hence 2a + (p-1) d = 2q/p ..1

and q th term a + (q-1) p and sum of 1st q terms = ( 2a + (p-1)q)* (p/2) = p

hence 2a + (q-1) d = 2p/q ..2

subtract (2) from (1)
(p-q) d = 2(q/p-p/q) = 2(q^2-p^2)/(pq)

so d = -2 (p+q)/(pq)

now sum of 1st (p+q) terms

now (p+q)th term = ( a + (p+q-1) d)

and sum of its first (p + q) terms = ( 2a + (p+q -1) d) (p+q)/2
= ( 2a + (p-1) d + qd) (p+q)/2
= ( 2 q/p + qd) (p+ q)/2 as 2a + (p-1)d = 2q/p
= ( 2q/p + q (-2(p+q)/pq) * (p+q)/2 putting value of d
= ( 2q/p - 2 - 2q/p) * (p+q)/2
= - (p+ q)