If the geometric mean of a and b is `(a^(n+1)+b^(n+1))/(a^(n)+b^(n))` then n = ?

Options

A. `-(1)/(2)`
B. `(1)/(2)`
C. `1`
D. None of these

Explanation

To solve the problem, we need to find the value of \( n \) given that the geometric mean of \( a \) and \( b \) is equal to \( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \).

### Step-by-Step Solution:

1. **Understand the Geometric Mean**:
The geometric mean of two numbers \( a \) and \( b \) is given by:
\[ \text{Geometric Mean} = \sqrt{ab} \]

2. **Set Up the Equation**:
According to the problem, we have:
\[ \sqrt{ab} = \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \]

3. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ ab = \left(\frac{a^{n+1} + b^{n+1}}{a^n + b^n}\right)^2 \]

4. **Cross Multiply**:
Cross multiplying gives us:
\[ ab(a^n + b^n)^2 = (a^{n+1} + b^{n+1})^2 \]

5. **Expand Both Sides**:
- Left Side:
\[ ab(a^{2n} + 2a^n b^n + b^{2n}) = a^{2n+1}b + ab^{2n+1} + 2a^{n+1}b^{n+1} \]
- Right Side:
\[ a^{2(n+1)} + 2a^{n+1}b^{n+1} + b^{2(n+1)} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ a^{2n+1}b + ab^{2n+1} + 2a^{n+1}b^{n+1} = a^{2n+2} + 2a^{n+1}b^{n+1} + b^{2n+2} \]
Cancel out \( 2a^{n+1}b^{n+1} \) from both sides:
\[ a^{2n+1}b + ab^{2n+1} = a^{2n+2} + b^{2n+2} \]

7. **Factor Out Common Terms**:
Factor out \( ab \) from the left side:
\[ ab(a^{2n} + b^{2n}) = a^{2n+2} + b^{2n+2} \]

8. **Divide Both Sides by \( ab \)**:
This gives:
\[ a^{2n-1} + b^{2n-1} = a^{2n+1} + b^{2n+1} \]

9. **Set Up the Equation**:
From this, we can equate the powers:
\[ a^{2n-1} = b^{2n-1} \]
This implies:
\[ \left(\frac{a}{b}\right)^{2n-1} = 1 \]

10. **Solve for \( n \)**:
Therefore, we have:
\[ 2n - 1 = 0 \implies 2n = 1 \implies n = \frac{1}{2} \]

### Final Answer:
Thus, the value of \( n \) is:
\[ n = -\frac{1}{2} \]