Question

If sum of first $p$ terms is $q$ and sum of first $q$ terms is $p$ then what is the sum of first $p+q$ terms

Answer 1

sum of $p$ terms $=\frac{p}{2}(2a+(p-1\left)d\right)$

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

sum of $q$ terms $=\frac{q}{2}(2a+(q-1\left)d\right)$

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

from eq (1) and (2) value of a and d can be calculated,

$a=\frac{q}{p}$

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

sum of $p+q$ terms $=\frac{p+q}{2}(2a+(p+q-1\left)d\right)$

after putting values of a and d we will get

$sum=-(p+q)$