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.

Answer

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

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

### Step 1: Write the formula for the geometric mean
The geometric mean of two numbers \( a \) and \( b \) is defined as:

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

### Step 2: Set the two expressions equal
We can set the two expressions for the geometric mean equal to each other:

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

### Step 3: Square both sides
To eliminate the square root, we square both sides:

\[ ab = \left(\frac{a^n + b^n}{a^{n-1} + b^{n-1}}\right)^2 \]

### Step 4: Cross-multiply
Cross-multiplying gives us:

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

### Step 5: Expand both sides
Now we expand both sides:

1. Left-hand side:
\[ ab(a^{2(n-1)} + 2a^{n-1}b^{n-1} + b^{2(n-1)}) = a^{n+1}b + 2ab^{n} + ab^{n+1} \]

2. Right-hand side:
\[ (a^n + b^n)^2 = a^{2n} + 2a^nb^n + b^{2n} \]

### Step 6: Equate and simplify
Now we equate the two sides:

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

### Step 7: Analyze the powers of \( a \) and \( b \)
To find \( n \), we can analyze the powers of \( a \) and \( b \) on both sides.

### Step 8: Set coefficients equal
From the equation, we can set the coefficients of \( a \) and \( b \) equal. This leads us to find that:

\[ n = \frac{1}{2} \]

### Conclusion
Thus, the value of \( n \) is:

\[ \boxed{\frac{1}{2}} \]