Question

Four geometric means are inserted between the numbers $2^{11}-1$ and $2^{11}+1$. The product of these geometric means is

• A. $2^{44}-2^{23}+1$
• B. $2^{44}-2^{22}+1$
• C. $2^{22}-2^{11}+1$
• D. $2^{22}-2^{12}+1$

Answer 1

Answer: (A)

Let $a,ar,a{r}^2,a{r}^3,a{r}^4,a{r}^5$ be the $6$ terms of the $G.P$. having $4G.M.’s$ between $a$ and $a{r}^5$.
We have $a=2^{11}-1$ and
$a{r}^5=2^{11}+1$
$\therefore$ Product of geometric means $=\left(ar\right)\left(a{r}^2\right)\left(a{r}^3\right)\left(a{r}^4\right)$
$=a^4{r}^{10}$
$={\left(a^2{r}^5\right)}^2$
$={\left\{\left(a\right)\left(a{r}^5\right)\right\}}^2$
$={\left\{\left(2^{11}-1\right)\left(2^{11}+1\right)\right\}}^2$
$={\left(2^{22}-1\right)}^2$
$=2^{44}-2^{23}+1$.