<p>For which of the following values of <i>x</i>, 5<sup>th</sup> term is the numerically greatest term in the expansion of \((1 + x/3)^{10}\):</p>
Step-by-Step Solution
Key Concept: The numerically greatest term in a binomial expansion occurs when the ratio of consecutive terms transitions from being greater than 1 to less than 1. For (1 + x/3)^10, find where T_r/T_(r-1) ≤ 1 and T_(r+1)/T_r ≤ 1 simultaneously, then identify which x values satisfy this for r = 5.
<p><strong>Step 1:</strong> For expansion of (1 + x/3)^10, the general term is T_(r+1) = C(10,r)(x/3)^r.</p><p><strong>Step 2:</strong> For T_5 to be numerically greatest, we need: |T_5/T_4| ≥ 1 AND |T_6/T_5| ≤ 1.</p><p><strong>Step 3:</strong> Ratio T_(r+1)/T_r = |x(10-r)|/|3(r+1)|. For 5th term (r=4): T_5/T_4 = |6x|/15 and T_6/T_5 = |5x|/18.</p><p><strong>Step 4:</strong> From T_5/T_4 ≥ 1: |6x|/15 ≥ 1 ⟹ |x| ≥ 2.5</p><p><strong>Step 5:</strong> From T_6/T_5 ≤ 1: |5x|/18 ≤ 1 ⟹ |x| ≤ 3.6</p><p><strong>Step 6:</strong> Therefore 2.5 ≤ |x| ≤ 3.6, which means x ∈ [-3.6, -2.5] ∪ [2.5, 3.6].</p><p>∴ Answer: B</p>
Correct Answer: B