Question

If the $p^{th}$ term of an AP is $q$ and the $q^{th}$ term is $p$, then $n^{th}$ term of the A.P is:

• A. $p+q-n$
• B. $p-q-n$
• C. $p+q$
• D. $p-n$

Answer 1

The correct option is A $p+q-n$

$a_p=a+(p-1)d=q...\left(1\right)a_q=a+(q-1)d=p...\left(2\right)$

$\left(2\right)-\left(1\right)⟹p-q=d\left[\right(q-1)-(p-1\left)\right]$
$=d(q-p)$

$⟹d=\frac{p-q}{q-p}=-1$

Substituting the value of $p$ in $\left(1\right)⟹a=p+q-1$

Substitute the values of $a$ and $d$ in
$a_n=a+(n-1)d$
$⟹a_n=p+q-1+(n-1)-1$
$a_n=p+q-n$