Question

In an AP given l = 28, S = 144, and there are total 9 terms. Find a.

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

Answer 1

Given that, l = 28, S = 144 and there are total of 9 terms.

`S_n = n/2(a+1)`

`144 = 9/2(a+28)`

⇒ a + 28 = `(144 xx 2)/9`

⇒ a = 16 × 2

⇒ a = 32

⇒ a = 32 - 28

⇒ a = 4

Answer 2

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 (a₉) = 28

Sum of n terms (S_n) = 144

Number of terms (n) = 9

Now,

a₉ = 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