Question

In a H.P. $p^{th}$ term is $q$ and $q^{th}$ term is $p$. Then the $(pq{)}^{th}$ term is

• A. Zero
• B. $\frac{p+q}{pq}$
• C. $1$
• D. $\frac{pq}{p+q}$

Answer 1

The correct option is C $1$

Solution:

Given that: $t_p=q,t_q=p$

To find: $t_{pq}=?$

Solution:

Let, $a$ be first term and $d$ be common difference of an $A.P.$

$t_p=q⟹\frac{1}{a+(p-1)d}=q⟹a+(p-1)d=\frac{1}{q}............\left(i\right)$ and

$t_q=p⟹\frac{1}{a+(q-1)d}=p⟹a+(q-1)d=\frac{1}{p}.................\left(ii\right)$

Subtracting eqs. (ii) from (i), we get

$(p-q)d=\frac{1}{q}-\frac{1}{p}⟹(p-q)d=\frac{p-q}{pq}$

or, $d=\frac{1}{pq}$

Putting value of $d$ in eqn(i), we get

$a=\frac{1}{pq}$

Now,

$t_{pq}=\frac{1}{a+(pq-1)d}=\frac{1}{\frac{1}{pq}+(pq-1)\frac{1}{pq}}=\frac{pq}{1+pq-1}=\frac{pq}{pq}=1$

Hence, C is the correct option.