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 \( n \) such that the geometric mean of \( a \) and \( b \) is given by the expression:

\[ \text{GM} = \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \]

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

\[ \text{GM} = \sqrt{ab} \]

### Step 2: Set up the equation
According to the problem, we have:

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

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

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

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

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

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

Left-hand side:

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

Right-hand side:

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

### Step 6: Set the equation
Now we have:

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

### Step 7: Rearranging the equation
Rearranging gives us:

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

### Step 8: Factor out common terms
Factoring out common terms, we can analyze the powers of \( a \) and \( b \) to find \( n \).

### Step 9: Equate the coefficients
By equating the coefficients of \( a \) and \( b \), we can find the value of \( n \).

### Step 10: Solve for \( n \)
After simplification, we find that:

\[ n = 1 \]

Thus, the value of \( n \) is \( 1 \).