Question

If $pth$ term of an arithmetic progression is $q$and the $qth$ term is $p$, then $10th$ term is:

• A. $p-q+10$
• B. $p+q+11$
• C. $p+q-9$
• D. $p+q-10$

Answer 1

The correct option is D

$p+q-10$

Explanation for the correct option.

Step 1: Find the common difference.

Given, $a_p=q$ & $a_q=p$.

We know that nth term in an A.P is given by: $a_n=a+(n-1)d$, where $a,n,d$ are first term, number of terms & common difference respectively.

$a_p=a+(p-1)d\mathrm{and}{\mathrm{a}}_{\mathrm{q}}=\mathrm{a}+(\mathrm{q}-1)\mathrm{d}$

$$\begin{gathered} q=a+(p-1)d....\left(1\right) \\ p=\mathrm{a}+(\mathrm{q}-1)\mathrm{d}...\left(2\right) \end{gathered}$$

Subtract equation (1) from (2) we get:

$\begin{array}{rcl}p-q& =& \left(q-1-p+1\right)d\\ p-q& =& -(p-q)d\\ d& =& -1\end{array}$

Step 2: Find 10th term

First, find the first term for this substitute $d=-1$in equation (1) we get;

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

Therefore, the 10th term is:

$\begin{array}{rcl}{a}_{10}& =& a+\left(10-1\right)d\\ a_{10}& =& q+p-1-9\\ a_{10}& =& q+p-10\end{array}$

So, the 10th term is $p+q-10$.

Hence, option D is correct.