Question

If twice the ${11}^{\mathrm{th}}$ term of an A.P is equal to $7$ times of its $21\mathrm{st}$ term, then its $25\mathrm{th}$ term is equal to

• A. $24$
• B. $120$
• C. $0$
• D. None of these

Answer 1

The correct option is C

$0$

Explanation for the correct answer:

Finding the value of $\mathbf{25}\mathbf{th}$ term:

Let the first term of A.P be $\mathrm{a}$ and common difference be $\mathrm{d}$.

${11}^{\mathrm{th}}$ term of A.P$=\mathrm{a}+10\mathrm{d}$ $\left[\because t_n=a+\left(n-1\right)d\right]$

${21}^{\mathrm{st}}$ term of A.P$=\mathrm{a}+20\mathrm{d}$

Given that,

$$\begin{gathered} 2\left(\mathrm{a}+10\mathrm{d}\right)=7\left(\mathrm{a}+20\mathrm{d}\right) \\ \Rightarrow 2\mathrm{a}+20\mathrm{d}=7\mathrm{a}+140\mathrm{d} \\ \Rightarrow 5\mathrm{a}+120\mathrm{d}=0 \\ \Rightarrow \mathrm{a}+24\mathrm{d}=0 \end{gathered}$$

And the $25\mathrm{th}$ term is $\mathrm{a}+24\mathrm{d}$. So the $25\mathrm{th}$ term is $0$.

Hence, the correct option is C.