Question

If the sum of the first $100$ terms of an $AP$ is $-1$ and the sum of even terms lying in first $100$ terms is $1,$ then which of the following is not true?

• A. Common difference of the sequence is $\frac{3}{50}$
• B. First term of the sequence is $\frac{-149}{50}$
• C. $100^{\text {th }}$ term $=\frac{74}{25}$
• D. None of these

Answer 1

Answer: (D)

$x_{1}+x_{2}+x_{3}+\ldots+x_{100}=\frac{100}{2}\left(x_{1}+x_{100}\right)=-1$
$\Rightarrow x_{1}+x_{100}=-\frac{1}{50}$
$x_{2}+x_{4}+\ldots+x_{100}=\frac{50}{2}\left(x_{1}+d+x_{100}\right)=1$
$\Rightarrow x_{1}+x_{100}+d=\frac{1}{25}$
$\Rightarrow d=\frac{3}{50}$
$x_{1}+x_{1}+99 d=\frac{-1}{50}$
$\Rightarrow x_{1} =\frac{-149}{50} $
$x_{100} =x_{1}+99 d $
$=\frac{-149}{50}+99 \times \frac{3}{50} $
$=\frac{74}{25}$