Question

In an AP, the $p^{th}$ term is q, and the $q^{th}$ term is p. What is the $n^{th}$ term? ( Given $p\ne q$)

Answer 1

Let a and d be the first term and the common difference of the AP. We have

$T_p=a+(p-1)d=q$ ...... (i)

$T_q=a+(q-1)d=p$ ...... (ii)

By, (i) - (ii), we have

$T_p-T_q=(p-q)d=q-p$

$\Rightarrow d=-1$

Using this value in (i) gives

$a=p+q-1$

Now, the $n^{th}$ term is

$T_n=a+(n-1)d$

$=(p+q-1)+(n-1)(-1)$

$=p+q-n$