#### Question

Let there be an A.P. with the first term ‘a’, common difference 'd’. If a denotes its nth term and Sn the sum of first n terms, find.

a, if a_{n} = 28, S_{n} = 144 and n= 9.

#### Solution

Here, we have an A.P. whose *n*^{th} term (*a*_{n}), the sum of first *n* terms (*S*_{n}) and the number of terms (*n*) are given. We need to find first term (*a*).

Here,

Last term (`a_9`) = 28

Sum of *n* terms (*S*_{n}) = 144

Number of terms (*n*) = 9

Now,

`a_9 = a + 8d`

28 = a + 8d .....(1)

Also, using the following formula for the sum of *n* terms of an A.P

`S_n = n/2[2a + (n - 1)d]`

Where; *a* = first term for the given A.P.

*d* = common difference of the given A.P.

*n *= number of terms

So, using the formula for *n* = 9, we get,

`S_8 = 9/2 [2a + (9 -1 )(d)]`

144(2) = [2a + 8d]

288 = 18a + 72d ....(2)

Multiplying (1) by 9, we get

9a + 72d = 252 ....(3)

Further substracting (3) from (2) we get

9a = 36

`a = 36/9`

a = 4

Therefore, the first term of the given A.P. is a = 4