Question

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

Answer 1

$a=54$

$d=-3$

$S_n=513$

$S_n=\frac{n}{2}(2a+(n-1\left)d\right)$

$513=\frac{n}{2}(108+3-3n)$

$1026=111n-3n^2$

$n^2-37n+342=0$

$(n-18)(n-19)=0$

on solving, we get

$n=18,n=19$

We are getting here 2 values, on checking we can conclude that as the ${19}^{th}$ term is 0 so adding it in the series doesn't make a difference.