<p>If \(\dfrac{a^n + b^n}{a^{n-1} + b^{n-1}}\) is the A.M. between <em>a</em> and <em>b</em>, then find the value of <em>n</em>.</p>
Step-by-Step Solution
Key Concept: The arithmetic mean of a and b equals (a+b)/2. Set the given expression equal to this and use algebraic manipulation to find when the equality holds for all values of a and b.
<p><strong>Step 1:</strong> If the given expression is the A.M. between a and b, then:</p><p>$$\frac{a^n + b^n}{a^{n-1} + b^{n-1}} = \frac{a + b}{2}$$</p><p><strong>Step 2:</strong> Cross-multiply:</p><p>$$2(a^n + b^n) = (a + b)(a^{n-1} + b^{n-1})$$</p><p><strong>Step 3:</strong> Expand the right side:</p><p>$$2a^n + 2b^n = a^n + ab^{n-1} + ba^{n-1} + b^n$$</p><p><strong>Step 4:</strong> Simplify:</p><p>$$a^n + b^n = ab^{n-1} + a^{n-1}b$$</p><p>$$a^n + b^n = ab(b^{n-2} + a^{n-2})$$</p><p><strong>Step 5:</strong> For this to hold for all a and b, test n = 1:</p><p>$$a^1 + b^1 = ab(b^{-1} + a^{-1})$$</p><p>$$a + b = ab\left(\frac{a+b}{ab}\right)$$</p><p>$$a + b = a + b$$ ✓</p><p>∴ <strong>Answer: n = 1</strong></p>
Correct Answer: 1