<p>Find \(a\) if the 17th and 18th terms of the expansion \((2 + a)^{50}\) are equal.</p>
Step-by-Step Solution
Key Concept: In a binomial expansion, consecutive terms are equal when their ratios equal 1. Set up the equation T₁₇ = T₁₈ using the general term formula and the ratio of consecutive terms will directly give you the unknown.
<p><strong>Step 1:</strong> Use the ratio of consecutive terms. If T₁₇ = T₁₈, then T₁₈/T₁₇ = 1.</p><p><strong>Step 2:</strong> For expansion of (2 + a)⁵⁰, the general term is T_{r+1} = C(50,r) · 2^{50-r} · a^r</p><p>So: T₁₈/T₁₇ = [C(50,17) · 2^{33} · a^{17}] / [C(50,16) · 2^{34} · a^{16}]</p><p><strong>Step 3:</strong> Simplify:</p><p>T₁₈/T₁₇ = [C(50,17)/C(50,16)] · [a/2] = [(50-16)/(17)] · [a/2] = [34/17] · [a/2] = a</p><p><strong>Step 4:</strong> Since T₁₈/T₁₇ = 1:</p><p>a = 1</p><p>∴ Answer: <strong>a = 1</strong></p>
Correct Answer: 1