Find the second term and nth term of an A.P.

In the given problem, we are given 6th and 8th term of an A.P.

We need to find the 2nd and nth term

Here, let us take the first term as a and the common difference as d

We are given,

$a_{6}=12$

$a_{5}=22$

Now, we will find $a_{6}$ and $a_{8}$ using the formula $a_{e}=a+(n-1) d$

So,

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

$12=a+5 d$ .......(1)

Also,

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

$22=a+7 d$.........(2)

So, to solve for a and d

On subtracting (1) from (2), we get

$22-12=(a+7 d)-(a+5 d)$

$10=a+7 d-a-5 d$

$10=2 d$

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

$d=5$ ........(3)

Substituting (3) in (1), we get

$12=a+5(5)$

$a=12-25$

$a=-13$

Thus,

$a=-13$

$d=5$

So, for the 2nd term (n = 2),

$a_{2}=-13+(2-1) 5$

$=-13+(1) 5$

$=-13+5$

$=-8$

For the nth term,

$a_{e}=-13+(n-1) 5$

$=-13+5 n-5$

$=-18+5 n$

Therefore, $a_{2}=-8, a_{n}=5 n-18$