# How many terms of the series 54, 51, 48, …. be taken so that their sum is 513 ? Explain the double answer - Mathematics

Sum

How many terms of the series 54, 51, 48, …. be taken so that their sum is 513 ? Explain the double answer

#### Solution

∵ a = 54, d = – 3 and S_n = 513

⇒ \frac { n }{ 2 } [2a + (n – 1) d] = 513

⇒ \frac { n }{ 2 } [108 + (n – 1) × – 3] = 513

⇒ n2 – 37n + 342 = 0

⇒ (n – 18) (n – 19) = 0 ⇒ n = 18 or 19

Here, the common difference is negative, So, 19th term is a19 = 54 + (19– 1) × – 3 = 0.

Thus, the sum of 18 terms as well as that of 19 terms is 513

Concept: Sum of First n Terms of an AP
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