Question

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

• A. only $18$ number of terms
• B. only $17$ number of terms
• C. only $19$ number of terms
• D. both $18$ and $19$ number of terms

Answer 1

The correct option is D both $18$ and $19$ number of terms

The first term of an A.P. is $54$ and common difference is $-3.$
So, sum of $n$ terms is $\frac{n(2a+nd-d)}{2}$.
Here, $\frac{n(108-3n+3)}{2}=513.$
$\Rightarrow 111n-3n^2=1026$
$\Rightarrow n^2-37n+342=0$
$\Rightarrow n=18,19.$
There are two possible values for n since the ${19}^{th}$ term is $0$ and thus the sum remains unchanged.