Question

Find the common difference of an A.P whose first term is unity and whose second, tenth and thirty fourth terms are in G.P

• A. $\frac{1}{5}$
• B. $\frac{1}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{9}$

Answer 1

The correct option is B

$\frac{1}{3}$

Finding common difference:

Let assume an A.P with terms $a,a+d,a+2d,a+3d,\dots$ and so on.

First term is $a$

$N^{th}$ term $\left(T_n\right)=a+\left(n-1\right)d$

$$\begin{gathered} T_2=a+d \\ {T}_{10}=a+9d \\ {T}_{34}=a+33d \end{gathered}$$

Given, $T_2,T_{10},T_{34}$ are in G.P.

If $f,g,h$ is in G.P. then, $g^2=fh$.

So, ${T_2}^2=T_{10}\times T_{34}$

$$\begin{gathered} \Rightarrow {\left(a+9d\right)}^2=\left(a+d\right)\times (a+33d) \\ \Rightarrow a^2+81d^2+18ad=a^2+33d^2+34ad \\ \Rightarrow 48d^2=16d \\ \Rightarrow d=\frac{1}{3} \end{gathered}$$

Hence, common difference is $\frac{1}{3}$.

Thus, the correct answer is option (B).