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Find the sum of first 22 terms of the AP whose first term is 8 and common difference is -5​

Answer»

In the given PROBLEM, we need to find the number of terms of an A.P. Let us take the number of terms as n.Here, we are given that,38a = 22d = -4S_n= 6SO, as we know the formula for the sum of n terms of an A.P. is given by,So, as we know the formula for the sum of n terms of an A.P. is given by,'S_n= n/2 [2a + (n - 1)d]38Where; a = first term for the given A.P.d = common difference of the given A.P.n = number of termsSo, using the formula we get,A_n= n/2 [2(22) + (n - 1)(-4)]3864 = n/2[44 - 4 n + 4] =64(2) = n(48 - 4 n) =128 = 48n - 4n^2Further rearranging the terms, we get a quadratic equation,4n^2 - 48n + 128 = 0ON taking 4 common we get'n^2 - 12n + 32 = 0Further, on solving the equation for n by splitting the MIDDLE term, we get,'n^2 - 12n + 32 = 0`n^2 - 8n -4n + 32 = 0n(n 8) - 4(n - 8) = 0(n - 8)(n 4) = 038So, we get(n - 8) = 0n = 8Also(n - 4) = 0n = 4


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