Question

If $n$ geometric means be inserted between $a$ and $b$then the $n ^ { t h }$ geometric mean will be.

• A. $a \left( \frac { b } { a } \right) ^ { \frac { n } { n - 1 } }$
• B. $a \left( \frac { b } { a } \right) ^ { \frac { n - 1 } { n } }$
• C. $a \left( \frac { b } { a } \right) ^ { \frac { n } { n + 1 } }$
• D. $a \left( \frac { b } { a } \right) ^ { \frac { 1 } { n } }$

Answer 1

Answer: (C)

$a \left( \frac { b } { a } \right) ^ { \frac { n } { n + 1 } }$

Answer 2

If $n$ geometric means $g _ { 1 } , g _ { 2 } \ldots \ldots . . g _ { n }$ are to be inserted between two positive real numbers $a$ and $b$, then are in G.P. Then

So $b = a r ^ { n + 1 } \Rightarrow r = \left( \frac { b } { a } \right) ^ { 1 / ( n + 1 ) }$

Now $n ^ { t h }$ geometric mean $\left( g _ { n } \right) = a r ^ { n } = a \left( \frac { b } { a } \right) ^ { n / ( n + 1 ) }$.

Aliter : As we have the $m ^ { t h }$ G.M. is given by

$G _ { m } = a \left( \frac { b } { a } \right) ^ { \frac { m } { n + 1 } }$

Now replace $m$ by $n$ we get the required result.