Question

If the sum of first p terms, first q terms and first r terms of an A.P. be a, b and c respectively,then $\frac{a}{p}$(q - r) + $\frac{b}{q}$(r - p) + $\frac{c}{r}$(p - q) is equal to :

Answer 1

$S_p$ = $\frac{p}{2}$[2A + (p - 1)d] = a

$\therefore$ $\frac{2a}{p}$ = 2A + (p - 1)d .......(i)

$\therefore$ $\frac{2b}{q}$ = 2A + (q - 1)d .......(ii)

$\therefore$ $\frac{2c}{r}$ = 2A + (r - 1)d .......(iii)

Multiplying (i), (ii) and (iii) by q-r, r-p and p-q respectively and add

$\therefore$ $\sum$ $\frac{a}{p}$(q-r) = 0