Question

How many terms of the series $54+51+48+45+.......$ must be taken to make $513$?

• A. only $n=18$
• B. only $n=19$
• C. only $n=17$
• D. both $n=18,19$

Answer 1

The correct option is D both $n=18,19$

Let the sum of $n$ terms of the series be $513$
$\because S_n=\frac{n}{2}\{2a+(n-1\left)d\right\}$
$\therefore 513=\frac{n}{2}\{2.54+(n-1\left)\right(-3\left)\right\}$
$\Rightarrow 171=\frac{n}{2}\{2.18-(n-1\left)\right\}$
$\Rightarrow 342=n\{36-n+1\}$
$\Rightarrow 342=n\{37-n\}$
$\Rightarrow n^2-37n+342=0$
$(n-19)(n-18)=0$
$\therefore n=18,19$
Explanation of double answer:
$T_{18}=18$th term $=54+(18-1)(-3)=3$
$T_{19}=19$th term $=54+(19-1)(-3)=0$
Since $19$th term is zero, sum of 18 terms and 19 terms are both $513$.