If the sum of n terms of an A.P. is 3n2 + 5n then which of its terms is 164?

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Here, the sum of first n terms is given by the expression, Sn = 3n2 + 5n We need to find which term of the A.P. is 164. Let us take 164 as the nth term So we know that the nthterm of an A.P. is given by, an = Sn - Sn-1 So, 164 = Sn - Sn-1 164 = 3n2 + 5n - [3(n-1)2 +5(n-1) ] Using the property, ( a - b)2 = a2 + b2 - 2ab We get, 164 = 3n2 + 5n - [3(n2 + 1 - 2n) + 5 ( n-1)] 164 = 3n2 + 5n - [3n2 + 3 - 6n + 5n - 5] 164 = 3n2 + 5n -(3n2 - n - 2) 164 = 3n2 + 5n - 3n2 + n + 2 164 = 6n + 2 Further solving for n, we get 6n = 164 - 2 n = $\frac{162}{6}$ n = 27 Therefore, 164 is the 27th term of the given A.P.

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