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

Question:

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

Solution:

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

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