Question

If $100$ times the $100$th term of an $AP$ with non-zero common difference equals to $50$ times its $50$th term, then the $150$th term of an $AP$ is

• A. $-150$
• B. $150$ times its $50$th term
• C. $150$
• D. Zero

Answer 1

The correct option is D

Zero

Explanation for the correct option:

Find the value of $T_{150}$ of given $AP$:

Let $d$ be the common difference and $a$ be the first term of given $AP$.

$T_{150}=a+149d$ …(1) $\left[\mathbf{\because }{\mathit{T}}_{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}\mathit{a}\mathbf{}\mathbf{+}\mathbf{}\left(n-1\right)\mathit{d}\right]$

Given, $100\left(T_{100}\right)=50\left(T_{50}\right)$

$\Rightarrow$ $100(a+99d)=50(a+49d)$

$\Rightarrow$ $2(a+99d)=(a+49d)$

$\Rightarrow$ $2a+198d-a-49d=0$

$\Rightarrow$ $a+149d=0$ …(2)

From (1) and (2), we get

${\mathit{T}}_{\mathbf{150}}=\mathbf{0}$

Hence, Option ‘D’ is Correct