Question

The common difference, last term, and sum of terms of an A.P. are 4, 31, and 136, respectively. Find the number of terms.

Answer 1

Given:

d = 4

Last term = 31

S_n = 136

a_l = a + (n − 1)d

31 = a + (n − 1)4

31 = a + 4n − 4

31 + 4 = a + 4n

35 = a + 4n

a + 4n = 35

a = 35 − 4n

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

136 = `n/2(35 − 4n + 31`

136 × 2 = n(66 − 4n)

272 = 66n − 4n²

4n² − 66n + 272 = 0

Dividing by 2:

2n² − 33n + 136 = 0

2n² − 16n − 17n + 136 = 0

2n(n − 8) − 17(n − 8) = 0

(2n − 17) (n − 8) = 0

n = `17/2`, n = 8

n = 8.5, n = 8

The number of terms in the A.P. is 8.