The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Answer

N/a

Explanation

Let n be the number of terms and d be common difference of A.P.
`because` First term a = 11,
`therefore` Second term a + d = 11 + d,
Third term a + 2d = 11 + 2d
Fourth term a + 3d = 11 + 3d
Then the sum of first four terms
= 11 + (11 + d) + (11 + 2d) + (11 + 3d)
= 44 + 6 d
`therefore` 44 + 6d = 56
`rArr` 6d = 56 - 44 = 12
`rArr` d = 2
`because` Last 4 terms = (n - 3)th term, (n - 2)th term, (n - 1)th term, nth term
(n - 3)th term of series = a + (n - 3 - 1)d
= 11 + (n - 4)2 = 2n + 3
Then last 4 terms of the series
= (2n + 3), (2n + 5), (2n + 7) + (2n + 9)
Their sum = (2n + 3) + (2n + 5) + (2n + 7) + (2n + 9)
= 8n + 24 = 8 (n + 3)
But the sum of last four terms = 112
`therefore` 8(n + 3) = 112
`rArr` `n + 3 = (112)/(8) = 14`
`rArr` n = 14 - 3 = 11
Therefore, number of terms = 11