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 an = 28, Sn = 144 and n= 9.
Here, we have an A.P. whose nth term (an), the sum of first n terms (Sn) and the number of terms (n) are given. We need to find first term (a).
Last term (`a_9`) = 28
Sum of n terms (Sn) = 144
Number of terms (n) = 9
`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