Question

If $p$ times the $pth$ term of an A.P is equal to $q$ times the $qth$ term of an A.P. then $(p+q)th$ term is:

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A

$0$

Explanation for the correct option.

Step 1: Write $pth\&qth$ term

Given that, $p$ times the $pth$ term of an A.P is equal to $q$ times the $qth$ term of an A.P.

We know that nth term in an A.P is given by $a_n=a+(n-1)d$.

$$\begin{gathered} a_p=a+(p-1)d \\ {a}_q=a+(q-1)d \end{gathered}$$

Step 2: Find $(p+q)th$term

According to the question :

$\begin{array}{rcl}p(a+(p-1)d)& =& q(a+(q-1)d)\\ p(a+(p-1)d)-q(a+(q-1)d)& =& 0\\ a(p-q)+\left(p^2-q^2\right)d+(q-p)d& =& 0\\ a(p-q)+(p-q)(p+q)d+(q-p)d& =& 0\left[\because {\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{a}-\mathrm{b}\right)\right]\\ a(p-q)+(p-q)d(p+q-1)& =& 0\end{array}$

Dividing both sides by $(p-q)$ we have:

$a+(p+q-1)d=0$

Which is nothing but $\left(p+q\right)th$ the term A.P.

Hence, option A is correct.