Question

The value of $ n $, for which $ \frac{a^{n+1} + b^{n+1}}{a^n + b^n} $ is the A.M. between $ a $ and $ b $, is

• A. 0
• B. 1
• C. \( -\frac{1}{2} \)
• D. -1

Hint:

For expressions involving powers of \( a \) and \( b \), substitute different values of \( n \) and check for conditions that match the definition of A.M.

Answer 1

The Correct Option is B

Step 1: Understand the concept of Arithmetic Mean.
The Arithmetic Mean (A.M.) of two numbers \( a \) and \( b \) is given by: \[ A.M.
= \frac{a + b}{2} \] We are asked to find the value of \( n \) for which: \[ \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \] is the A.M.
between \( a \) and \( b \).
Step 2: Set up the equation.
For the given expression to be the A.M.
between \( a \) and \( b \), it should be equal to: \[ \frac{a + b}{2} \]
Step 3: Analyze the expression.
Let’s simplify the given expression for different values of \( n \).
The value of \( n \) that satisfies the equation is found to be 1, as this makes the expression equal to \( \frac{a + b}{2} \).
Step 4: Conclusion.
Therefore, the value of \( n \) for which the expression is the A.M.
between \( a \) and \( b \) is 1.