Question

If \[ \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \] is the arithmetic mean (A.M.) between \( a \) and \( b \), then the value of \( n \) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For sequences, always verify whether the given condition satisfies the definition of arithmetic mean. In this case, the form \( \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \) simplifies to the correct arithmetic mean when \( n = 1 \).

Answer 1

The Correct Option is A

We are given the expression: \[ \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \] and told that it represents the arithmetic mean (A.M.) of \( a \) and \( b \). Recall that the arithmetic mean of two numbers \( a \) and \( b \) is given by: \[ \frac{a + b}{2} \] We need to find the value of \( n \) for which the given expression matches the arithmetic mean. For \( n = 1 \), the expression becomes: \[ \frac{a^1 + b^1}{a^{1-1} + b^{1-1}} = \frac{a + b}{a + b} = 1 \] Thus, the expression simplifies to 1 when \( n = 1 \), which is the correct form for the arithmetic mean between \( a \) and \( b \). Therefore, the correct answer is \( n = 1 \).