Question

In an AP if the first term is 22 the common difference is -4 and the sum to nth term is 64 find n

Answer 1

Formula used :

$\boxed{\mathbf{S_{n} = \frac{n}{2}(2a + (n - 1)d)}}$

Given,
First term of an AP (a) = 22
Common difference (d) = - 4
$Sum \: of \: nth \: terms (S_{n})$ = 64
Number of terms (n) = (to be calculated)

$\underline{\underline{\huge\mathfrak{Solution}}}$

Putting the given values in the formula we get,

$64 = \frac{n}{2} (2(22) + (n - 1)( - 4)) \\ \\ 64 \times 2 = n(44 - 4n + 4) \\ \\ 128 = n(48 - 4n) \\ \\ 128 = 48n - 4 {n}^{2} \\ \\ - 4 {n}^{2} + 48n - 128 = 0 \\ \\ 4 {n}^{2} - 48n + 128 = 0 \\ \\ {n}^{2} - 12n + 32 = 0 \\ \\ {n}^{2} - 4n - 8n + 32 = 0 \\ \\ n(n - 4) - 8(n - 4) = 0 \\ \\ (n - 8) = 0 \: \: and \: \: (n - 4) = 0 \\ \\ \bf n = 8 \: or \: 4$

Let us find out the value of n that whether the value of n is 4 or 8.

Let us assume n = 4

$64 = \frac{4}{2} (2(22) + (4 - 1)( - 4)) \\ \\ 64 = 2(44 + (3)( - 4)) \\ \\ \frac{64}{2} = 44 - 12 \\ \\ 32 = 32$

Hence, the value of n = 12