Question

The ${10}^{th}$ term of an AP is 52 and ${16}^{th}$ term is 82. Find the ${32}^{nd}$ term and the general term.

Answer 1

Let a be the first term and d be the common difference of the given A.P. Let the A.P. be $a_1$, $a_2$, $a_3$..... $a_n$.....

It is given that

$a_{10}$ = 52 and $a_{16}$ = 82

$a+(10–1)d=52$ and $a+(16–1)d=82$

$a+9d=52$ .... (i)

and,

$a+15d=82$ .... (ii)

Subtracting equation (ii) from equation (i), we get

$–6d=–30$

$d=5$

Putting $d=5$ in equation (i), we get

$a+45=52$

$a=7$

$a_{32}=a+(32–1)d=7+31\times 5=162$

and,

$a_n=a+(n–1)d=7(n–1)\times 5=5n+2$.

Hence $a_{32}=162$ and $a_n=5n+2$.