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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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