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

Solution:

Given;

$a_{6}=12$

$\Rightarrow a+(6-1) d=12$

$\Rightarrow a+5 d=12$     ...(i)

$a_{8}=22$

$\Rightarrow a+(8-1) d=22$

$\Rightarrow a+7 d=22$     ...(ii)

Solving (i) and (ii), we get:

$2 d=10$

$\Rightarrow d=5$

Putting the value of $d$ in (i), we get:

$a+5 \times 5=12$

$\Rightarrow a=12-25=-13$

$\therefore a_{2}=a+(2-1) d=a+d=-13+5=-8$

Also, $a_{n}=a+(n-1) d$

$=-13+(n-1) 5$

$=-13+5 n-5$

$=5 n-18$