Question

If `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the G.M between a & b find the value of'n'?

Answer 1

`a=b` and `n` doesn't matter.

Explanation:

The geometric mean of `x` and `y` is given by `sqrt(xy)`.

Thus,

`sqrt(ab)=(a^n+b^n)/(a^(n-1)+b^(n-1))`

Cross multiplying yields:

`(ab)^(1/2)(a^(n-1)+b^(n-1))=a^n+b^n`

`a^(n-1/2)b^(1/2)+a^(1/2)b^(n-1/2)=a^n+b^n`

`a^nsqrt(b/a)+b^nsqrt(a/b)=a^n+b^n`

Comparing coefficients, we see that `sqrt(a/b)=1` and `sqrt(b/a)=1`. Thus, `a=b` and `n` is irrelevant.

Answer 2

`n=1/2`

Explanation:

Putting `n = 1/2` we find:

`(a^n+b^n)/(a^(n-1)+b^(n-1)) = (sqrt(a)+sqrt(b))/(1/sqrt(a)+1/sqrt(b))`

`color(white)((a^n+b^n)/(a^(n-1)+b^(n-1))) = (sqrt(a)+sqrt(b))/(((sqrt(a)+sqrt(b))/(sqrt(a)sqrt(b))))`

`color(white)((a^n+b^n)/(a^(n-1)+b^(n-1))) = sqrt(a)sqrt(b)`

`color(white)((a^n+b^n)/(a^(n-1)+b^(n-1))) = sqrt(ab)`

i.e. the geometric mean of `a` and `b`.

Answer 3

`n = 1/2`

Explanation:

Assuming that G.M stands for geometric mean, we have

`sqrt(a b) = (a^n + b^n)/(a^(n - 1) + b^(n - 1))` so squaring and simplifying we have

`ab = (a^(2n)+2 a^n b^n + b^(2n))/(a^(2n-2)+2a^(n-1)b^(n-1)+b^(2n-2))` or

`b a^(2n-1)+2a^nb^n+a b^(2n-1)=a^(2n)+2 a^n b^n + b^(2n)` or

`b/a a^(2n)+a/b b^(2n) = a^(2n)+b^(2n)` or

`(b-a)/a a^(2n)-(b-a)/b b^(2n) = 0`

Now assuming `a ne b`

`a^(2n)b-ab^(2n) = 0` or

`(a/b)^(2n)= a/b rArr 2n=1 rArr n = 1/2`