Question

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

Answer 1

$54,51,\mathrm{48......}$

Here $a=54,d=-3$

Let the sum of $n$ terms be $513$. Then,

$S_n=513$

$\frac{n}{2}[2a+(n-1\left)d\right]=513$

$\frac{n}{2}[108+(n-1)\times -3]=513$

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

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

$n=18$ or $19$.

Here, $d$ is negative. So, $19th$ term is given by

$a_{19}=54+(19-1)\times 3=0$

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