Question

In an A.P., if the first term is 22, the common difference is – 4 and the sum to n terms is 64, find n.

Answer 1

Given that,

a = 22, d = – 4 and S_n = 64

Let us consider the number of terms as n.

For sum of terms in an A.P, we know that

S_n = \(\frac{n}{2}\)[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,

⟹ S_n = \(\frac{n}{2}\)[2(22) + (n − 1)(−4)]

⟹ 64 = \(\frac{n}{2}\)[2(22) + (n − 1)(−4)]

⟹ 64(2) = n(48 – 4n)

⟹ 128 = 48n – 4n²

After rearranging the terms, we have a quadratic equation

4n² – 48n + 128 = 0,

n² – 12n + 32 = 0 [dividing by 4 on both sides]

n² – 12n + 32 = 0

Solving by factorisation method,

n² – 8n – 4n + 32 = 0

n ( n – 8 ) – 4 ( n – 8 ) = 0

(n – 8) (n – 4) = 0

So, we get n – 8 = 0 ⟹ n = 8

Or, n – 4 = 0 ⟹ n = 4

Hence, the number of terms can be either n = 4 or 8.