Question

If the $p^{th}$ term of an A.P is $\frac{1}{q}$ and $q^{th}$ term is $\frac{1}{p}$ then that prove that $p{q}^{th}$ term is one.

Answer 1

$a_p=a+(p-1)d=\frac{1}{q}\Rightarrow aq+pqd-dq=1$ ………….$\left(1\right)$

$a_q=a+(q-1)d=\frac{1}{p}\Rightarrow ap+pqd-dp=1$ ………….$\left(2\right)$

$\left(2\right)-\left(1\right)$

$\Rightarrow a(q-p)-d(q-p)=0$

$\Rightarrow a=d$

Substituting $a=d$ in equation $\left(1\right)$

$\Rightarrow pqd=1$

$a_{pq}=a+(pq-1)d$

$=d+(pq-1)d$

$=d(1+pq-1)$

$=pqd$

$=1$.

$a_{pq}=1$

Hence proved