If the G.M. of a and b is `(a^(n)+b^(n))/(a^(n-1)+b^(n-1))` then find the value of n.

Explanation

To solve the problem, we need to find the value of \( n \) such that the geometric mean (G.M.) of \( a \) and \( b \) is given by:

\[ \text{G.M.} = \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \]

We know that the geometric mean of two numbers \( a \) and \( b \) is given by:

\[ \text{G.M.} = \sqrt{ab} \]

So, we can set up the equation:

\[ \frac{a^n + b^n}{a^{n-1} + b^{n-1}} = \sqrt{ab} \]

### Step 1: Cross-multiply the equation

We can cross-multiply to eliminate the fraction:

\[ a^n + b^n = \sqrt{ab} \cdot (a^{n-1} + b^{n-1}) \]

### Step 2: Rewrite the right-hand side

We can express \( \sqrt{ab} \) as \( a^{1/2} b^{1/2} \):

\[ a^n + b^n = a^{1/2} b^{1/2} (a^{n-1} + b^{n-1}) \]

### Step 3: Expand the right-hand side

Distributing \( a^{1/2} b^{1/2} \):

\[ a^n + b^n = a^{n - 1/2} b^{1/2} + a^{1/2} b^{n - 1} \]

### Step 4: Rearrange the equation

Now, we can rearrange the equation:

\[ a^n - a^{n - 1/2} b^{1/2} + b^n - a^{1/2} b^{n - 1} = 0 \]

### Step 5: Factor the equation

We can factor the equation by grouping terms:

\[ a^{n - 1/2} (a^{1/2} - b^{1/2}) + b^{n - 1/2} (b^{1/2} - a^{1/2}) = 0 \]

### Step 6: Set each factor to zero

For the equation to hold true, either:

1. \( a^{1/2} - b^{1/2} = 0 \) which implies \( a = b \)
2. or \( a^{n - 1/2} + b^{n - 1/2} = 0 \) (which does not hold for positive \( a \) and \( b \))

### Step 7: Solve for \( n \)

If we assume \( a = b \), we can substitute \( b = ka \) (where \( k = 1 \)):

\[ \frac{a^n + a^n}{a^{n-1} + a^{n-1}} = \sqrt{a^2} \]

This simplifies to:

\[ \frac{2a^n}{2a^{n-1}} = a \]

This implies:

\[ a^{n - (n-1)} = 1 \implies a^{1} = 1 \]

Thus, \( n - 1 = 0 \) which gives us:

\[ n = 1 \]

So, the value of \( n \) is:

\[ \boxed{1} \]