If `(a^n+b^n)/(a^(n-1)+b^(n-1))` is the GM between a and b, then the value of n is

Options

A. 0
B. 1
C. `1//2`
D. none of these

Explanation

To solve the problem, we need to find the value of \( n \) such that

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

is the geometric mean (GM) of \( a \) and \( b \). The geometric mean of \( a \) and \( b \) is given by

\[ \sqrt{ab}. \]

### Step-by-Step Solution:

1. **Set up the equation**: We start with the equation given in the problem:
\[ \frac{a^n + b^n}{a^{n-1} + b^{n-1}} = \sqrt{ab}. \]

2. **Cross-multiply**: To eliminate the fraction, we can cross-multiply:
\[ a^n + b^n = \sqrt{ab} \cdot (a^{n-1} + b^{n-1}). \]

3. **Expand the right side**: We can expand the right-hand side:
\[ a^n + b^n = a^{n-1} \cdot \sqrt{ab} + b^{n-1} \cdot \sqrt{ab}. \]

4. **Rewrite the right side**: We can express \( \sqrt{ab} \) as \( a^{1/2} b^{1/2} \):
\[ a^n + b^n = a^{n-1} \cdot a^{1/2} b^{1/2} + b^{n-1} \cdot a^{1/2} b^{1/2}. \]
This simplifies to:
\[ a^n + b^n = a^{n - 1/2} b^{1/2} + b^{n - 1/2} a^{1/2}. \]

5. **Rearranging terms**: We can rearrange the equation:
\[ a^n + b^n - a^{n - 1/2} b^{1/2} - b^{n - 1/2} a^{1/2} = 0. \]

6. **Factor out common terms**: We can factor out \( a^{n - 1/2} \) and \( b^{n - 1/2} \):
\[ a^{n - 1/2} (a^{1/2} - b^{1/2}) + b^{n - 1/2} (b^{1/2} - a^{1/2}) = 0. \]

7. **Setting up the equality**: Since the expression equals zero, we can set the two factors equal to zero:
\[ a^{n - 1/2} (b^{1/2} - a^{1/2}) = 0. \]

8. **Solving for \( n \)**: Since \( a \) and \( b \) are positive, we focus on the exponent:
\[ n - \frac{1}{2} = 0 \implies n = \frac{1}{2}. \]

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

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