Question

If the $4^{th},7^{th}$ and ${10}^{th}$ terms of a G.P. be $a,b,c$ respectively, then the relation between $a,b,c$ is

• A. $b=\frac{\left(a+c\right)}{2}$
• B. $a^2=bc$
• C. $b^2=ac$
• D. $c^2=ab$

Answer 1

The correct option is C

$b^2=ac$

Finding the relation between $a,b,c$:

Step 1 : Find the terms using nth term of GP

The formula for the $n\text{th}$ of a G.P is given as

${\mathit{A}}_{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}{\mathit{r}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$, where $A_n$ is the $n\text{th}$ term, $A$ is the initial term, and $r$ is the common ratio.

The fourth term that is $a$ can be found as,

$a=A{r}^3\dots \left(i\right)$

The seventh term that is $b$ can be found as,

$b=A{r}^6\dots \left(ii\right)$

The tenth term that is $c$ can be found as,

$c=A{r}^9\dots \left(iii\right)$

Step 2: Find the relation between terms

If we multiply equations $\left(i\right)$and$\left(iii\right)$ we get

$\begin{array}{rcl}ac& =& A{r}^3\left(A{r}^9\right)\\ & =& A^2{r}^{12}\\ & =& {\left(A{r}^6\right)}^2\\ & =& b^2\left[\because b=A{r}^6\right]\end{array}$

From this, we can deduce that $b^2=ac$

Hence, the correct option is (C).