Question

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

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

Answer 1

The correct option is C

$b^2=ac$

Let the first term and the common ratio of a G.P. be $A$ and $r$ respectively.

We know that $n^{\text{th}}$ term of the G.P. is given by
$t_n=A{r}^{n-1}.$

Now, $t_4=a=A{r}^3,t_7=b=A{r}^6\text{and}{t}_{10}=A{r}^9.$

Now, $b^2=(A{r}^6{)}^2=A^2{r}^{12}$ and $ac=\left(A{r}^3\right)\left(A{r}^9\right)=A^2{r}^{12}=b^2.$

Hence, the relation $b^2=ac$ is true.