Let a and b be two positive unequal numbers , then `(a^(n+1)+b^(n+1))/(a^(n)+b^(n))` represents

Options

A. their A.M if n = 0
B. their G.M if `n = 1/2 `
C. their H.M if `n=-1`
D. none of these

Answer

To solve the problem, we need to evaluate the expression \((a^{n+1} + b^{n+1}) / (a^n + b^n)\) for different values of \(n\) and determine what it represents.

### Step-by-Step Solution:

1. **Evaluate for \(n = 0\)**:
- Substitute \(n = 0\) into the expression:
\[ \frac{a^{0+1} + b^{0+1}}{a^0 + b^0} = \frac{a^1 + b^1}{1 + 1} = \frac{a + b}{2} \]
- This is the arithmetic mean of \(a\) and \(b\).

2. **Evaluate for \(n = \frac{1}{2}\)**:
- Substitute \(n = \frac{1}{2}\) into the expression:
\[ \frac{a^{\frac{1}{2}+1} + b^{\frac{1}{2}+1}}{a^{\frac{1}{2}} + b^{\frac{1}{2}}} = \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{a^{\frac{1}{2}} + b^{\frac{1}{2}}} \]
- This expression does not simplify to the geometric mean \(\sqrt{ab}\). Therefore, it does not represent the geometric mean.

3. **Evaluate for \(n = -1\)**:
- Substitute \(n = -1\) into the expression:
\[ \frac{a^{-1+1} + b^{-1+1}}{a^{-1} + b^{-1}} = \frac{a^0 + b^0}{\frac{1}{a} + \frac{1}{b}} = \frac{1 + 1}{\frac{1}{a} + \frac{1}{b}} = \frac{2}{\frac{b + a}{ab}} = \frac{2ab}{a + b} \]
- This is the harmonic mean of \(a\) and \(b\).

4. **Conclusion**:
- From the evaluations:
- For \(n = 0\), it represents the arithmetic mean.
- For \(n = -1\), it represents the harmonic mean.
- For \(n = \frac{1}{2}\), it does not represent the geometric mean.

Thus, the final answer is that the expression represents the arithmetic mean when \(n = 0\) and the harmonic mean when \(n = -1\).