Find the second term and n^{th} term of an A.P. whose 6^{th} term is 12 and 8^{th} term is 22.

#### Solution

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

We need to find the 2^{nd} and n^{th} term

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

We are given,

`a_6 = 12`

`a_8 = 22`

Now we will find `a_6` and `a_8` using the formula `a_n = a + (n - 1)d`

So

`a_6 = a + (6 -1)d`

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

Also

`a_8 = a + (8 -1)d`

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

So to solve for a and d

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

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

10 = a + 7d - a - 5d

10 = 2d

`d = 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 2 nd term (n = 2)

`a_2 = -13 + (2 - 1)5`

= -13 + (1)5

= -13 + 5

= -8

For the nth term

`a_n = -13 + (n - 1)5`

= -13 + 5n - 5

= -18 + 5n

Therefore `a_2 = -8, a_n = 5n - 18`