Q: The pth term of an A.P. is a and qth term is b. then the sum of it’s (p + q) terms is

(A) $\large \frac{p+q}{2}[a+b+\frac{a-b}{p-q}]$

(B) $\large \frac{p+q}{2}[a-b-\frac{a-b}{p-q}]$

(C) $\large \frac{p+q}{2}[a+b+\frac{a+b}{p+q}]$

(D) None of these

Ans: (A).

Sol: Let x be the first term and d be the c . d of A.P.

a = x + (p – 1 ) d

b = x + ( q – 1)d

$\large d = \frac{a-b}{p-q} $

So , $\large x = a-\frac{(p-1)(a-b)}{p-q}$

$\large = \frac{pa-qa-pa + pb +a -b}{p-q}$

$\large = \frac{pb-qa+a-b}{p-q} $

Hence , $\large S_{p+q} = \frac{p+q}{2}[a+b+\frac{a-b}{p-q}] $