Question

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

• A. $\left(\frac{p+q}{p-q}\right)n$
• B. $\left(\frac{pq}{p-q}\right)n$
• C. $\frac{p-q}{n}$
• D. $p+q-n$

Answer 1

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

According to the question
$T_p=a+(p-1)d=q$ ...(1)
and

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

By subtracting (2) from (1), we get
$(p-1)d-(q-1)d=q-p$
$d((p-1)-(q-1))=(q-p)$
$d(p-1-q+1)=(q-p)$
$d(p-q)=(q-p)$
So $d=-1$ ...(3)

Putting value of $d$ in (1) , we get
$a+(p-1)-1=q$
$a-p+1=q$
So, $a=(q+p-1)$...(4)

Now, we know that
$t_n=a+(n-1)d$

By putting the value of $a$ and $d$, we get
$t_n=(p+q-1)+(n-1)-1$
$=p+q-1-n+1$
$t_n=p+q-n$