The $10^{t h}$ term of an A.P. is $52$ and $16^{t h}$ term is $82$ . Find the $32^{n d}$ term and the general term.

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Solution

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, $a + (16 - 1) D = 82$

$⟹ a + 9 d = 52$ ...(i)
and $a + 15 d = 82$ ...(ii)
Subtracting equation (ii) form equation (i), we get
$- 6 d = - 30 ⟹ d = 7$
Putting $d = 5$ in equation (i), we get

$a + 45 = 52 ⟹ a = 7$

$∴ a_{32} = a + (32 - 1) d = 7 + 31 \times 5162$

and, $a_{n} = a + (n - 1) d = 7 (n - 1) \times = 5 n + 2$

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

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