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# Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.

Given:

The 6th and 8th terms of an A.P. are 12 and 22 respectively.

To do:

We have to find the 2nd and nth terms.

Solution:

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

We know that,

nth term of an A.P. $a_n=a+(n-1)d$

Therefore,

$a_{6}=a+(6-1)d$

$12=a+5d$

$a=12-5d$......(i)

$a_{8}=a+(8-1)d$

$22=a+7d$

$22=(12-5d)+7d$ (From (i))

$22=12+2d$

$2d=22-12$

$2d=10$

$d=\frac{10}{2}$

$d=5$

Substituting $d=5$ in (i), we get,

$a=12-5(5)$

$a=12-25$

$a=-13$

2nd term of the A.P. $a_{2}=-13+(2-1)(5)$

$=-13+1(5)$

$=-13+5$

$=-8$

nth term $a_n=-13+(n-1)(5)$

$=-13+5n-5$

$=5n-18$

Hence, the 2nd and nth terms of the given A.P. are $-8$ and $5n-18$ respectively.

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