Question

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

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Given: $p^{th}$ term = q and $q^{th}$ term = p

To find: (p + q) terms is {-(p +q)}.

Step-by-step explanation:

According to question

$s_p=q$

$\frac{p}{2} 2a+(p-1)d}=q$

$2ap+p(p-1)d=2q-(i)$

$s_q=p$

$\frac{q}{2} {2a+(q-1)d}=p$

$2aq+q(q-1)d=2p-(ii)$

Subtract equation (ii) from equation (i), we get

$2a(p-q+p(p-1)-q(q-1)d=2p-2q$

$2a+(p+q-1)d=-2-(iii)$

$s_p+_q=\frac{p+q}{2} (2a+(p+q-1)d$

$s_p+_q=\frac{p+q}{2} (-2)$ (from (iii))

$s_p+_q=-(p+q)$ proved