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. 0

Answer 1

Hint: We assume the general terms of an arithmetic series. We find the formula for ${{t}_{n}}$, the ${{n}^{th}}$ term of the series. From that we express the ${{50}^{th}}$, ${{100}^{th}}$, ${{150}^{th}}$ term and use the relation to form the equation. We solve it and use the dependency in the ${{150}^{th}}$ term to find the solution.

Complete answer:
We express the arithmetic sequence in its general form.
We express the terms as ${{t}_{n}}$, the ${{n}^{th}}$ term of the series.
The first term be ${{t}_{1}}$ and the common difference be $d$ where $d={{t}_{2}}-{{t}_{1}}={{t}_{3}}-{{t}_{2}}={{t}_{4}}-{{t}_{3}}$.
We can express the general term ${{t}_{n}}$ based on the first term and the common difference.
The formula being ${{t}_{n}}={{t}_{1}}+\left( n-1 \right)d$.
Therefore, the ${{50}^{th}}$, ${{100}^{th}}$, ${{150}^{th}}$ terms will be ${{t}_{50}}={{t}_{1}}+49d$, ${{t}_{100}}={{t}_{1}}+99d$, ${{t}_{150}}={{t}_{1}}+149d$ respectively.
It is given that 100 times the ${{100}^{th}}$ term of an AP with non-zero common difference equals to 50 times its ${{50}^{th}}$ term.
Therefore, $100\left( {{t}_{1}}+99d \right)=50\left( {{t}_{1}}+49d \right)$. We simplify he equation.
$\begin{align}
  & 100\left( {{t}_{1}}+99d \right)=50\left( {{t}_{1}}+49d \right) \\
 & \Rightarrow 100{{t}_{1}}-50{{t}_{1}}=50d\left( 49-198 \right) \\
 & \Rightarrow 50{{t}_{1}}=-7450d \\
 & \Rightarrow {{t}_{1}}=-\dfrac{7450d}{50}=-149d \\
\end{align}$
Now we find the ${{150}^{th}}$ term ${{t}_{150}}={{t}_{1}}+149d$.
So, \[{{t}_{150}}={{t}_{1}}+149d=-149d+149d=0\].
And hence the correct answer is option D.

Note:
The sequence is an increasing sequence where the common difference is a positive number. We can evaluate from the multiplier of the terms to be equal. The common difference will never be calculated according to the difference of greater number from the lesser number.