Question

the pth term of an ap is a and the qth term of an ap is b. then prove that the sum of its (p+q) th term is p+q by 2 [ ( a+b) + (a-b) by (p-q) ] .

Answer 1

let 'x' be the first term of the ap

let 'd' be the common difference of the ap

given:

pth term= a

x+(p-1)d=a. ----(1)

and qth term= b

x+(q-1)d= b. ----(2)

add equation 1 and 2

2x+ (p+q-2)d=a+b

2x+(p+q-1)d -d= a+b

2x+ (p+q-1)d =a+b+d. -----(3)

Subtract equation 2 from 1

(p-q)d =a-b

d= a-b/p-q. ------(4)

now,

sum of (p+q)th term= p+q/2(2x+(p+q-1)d)

using equation 3

p+q/2(a+b+d)

using equation 4

= p+q/2(a+b+ a-b/p-q)

## hence proved ##