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