Question

If the sum of first $101$ terms of an A.P. is $0$ and If $1$ be the first term of the A.P. then the sum of next $100$ terms is

• A. $-101$
• B. $201$
• C. $-201$
• D. $-200$

Answer 1

The correct option is C

$-201$

Explanation for the correct option.

Step 1: Solve for the common difference $\left(d\right)$ .

Given: First term of A.P. $a=1$, sum of first $101$ terms is $S_{101}=0$.

We know that sum of first $n$ terms of A.P. is given by, $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

$$\begin{gathered} 0=\frac{101}{2}\left[2+100d\right] \\ \Rightarrow d=-\frac{2}{100} \\ \Rightarrow d=-\frac{1}{50} \end{gathered}$$

Step 2: Solve for the required sum.

Let $S$ is the sum of next $100$ terms of A.P.

$\begin{array}{rcl}S& =& S_{201}-S_{101}\\ & =& \frac{201}{2}\left[2+200\left(-\frac{1}{50}\right)\right]-0\left[\because S_{101}=0\right]\\ & =& 201\left[1-2\right]\\ & =& -201\end{array}$

So, the sum of next $100$ terms of A.P. is $-201$.

Hence option(C) is the correct option.