Question:
In an AP, the $p^{\text {th }}$ term is $q$ and $(p+q)^{\text {th }}$ term is 0 . Show that its $q^{\text {th }}$ term is $p$.
Solution:
Given: $p^{\text {th }}$ term is $q$ and $(p+q)^{\text {th }}$ term is 0 .
To prove: $q^{\text {th }}$ term is $p$.
$p^{\text {th }}$ term is given by
$q=a+(p-1) \times d \ldots \ldots$ equation 1
$(p+q)^{\text {th }}$ term is given by
$0=a+(p+q-1) \times d$
$0=a+(p-1) \times d+q \times d$
Using equation1
$0=q+q \times d$
$d=-1$
Put in equation1 we get
$a=q+p-1$
$q^{\text {th }}$ term is
$\Rightarrow q+p-1+(q-1) \times(-1)$
$\Rightarrow p$
Hence proved.