Question

Find the value of n so that may be the geometric mean between a and b .

Answer 1

Given that the geometric mean of G.P is a n+1 + b n+1 a n + b n .

We know that the geometric mean of a and b is ab .

So, according to the question,

a n+1 + b n+1 a n + b n = ab

Now, squaring on both side of above equation, we get

( a n+1 + b n+1 a n + b n ) 2 =( ab )

Solving the above equation.

a 2n+2 +2 a n+1 b n+1 + b 2n+2 =( ab )( a 2n +2 a n b n + b 2n ) a 2n+2 +2 a n+1 b n+1 + b 2n+2 = a 2n+1 b+2 a n+1 b n+1 +a b 2n+1 a 2n+2 + b 2n+2 = a 2n+1 b+a b 2n+1 a 2n+2 − a 2n+1 b=a b 2n+1 − b 2n+2

Further simplify above equation.

a 2n+1 ( a−b )= b 2n+1 ( a−b ) ( a b ) 2n+1 =1 ( a b ) 2n+1 = ( a b ) 0

Now, comparing the powers, we get

2n+1=0 n=− 1 2

Thus, the value of n is − 1 2 .