<p>In the binomial expansion of \(\left(\dfrac{x}{3}\right)^r\) in \(\left(2^{55-r}\right)\), two consecutive terms are equal. Find the value of \(r\).</p><p>Equivalently: In the expansion of \(\left(\dfrac{x}{3}\right)\) using \(^{32}C_r \cdot 2^{55-r}\), if two consecutive terms \(T_{r+1}\) and \(T_{r+2}\) are equal, then \(r\) equals:</p>
Step-by-Step Solution
Key Concept: Two consecutive binomial terms T_{r+1} and T_{r+2} are equal when their coefficient ratio equals the ratio of their variable parts. Use the condition C(n,r) = C(n,r+1) which occurs only when n is odd and r = (n-1)/2, or equate the general term expressions directly.
<p><strong>Step 1:</strong> In the expansion of (a+b)^n, the general term is T_{k+1} = C(n,k)·a^{n-k}·b^k</p><p><strong>Step 2:</strong> For the given expansion, T_{r+1} = C(55,r)·2^{55-r}·(x/3)^r and T_{r+2} = C(55,r+1)·2^{55-r-1}·(x/3)^{r+1}</p><p><strong>Step 3:</strong> Setting T_{r+1} = T_{r+2} (considering coefficients of like terms):</p><p>C(55,r)·2^{55-r} = C(55,r+1)·2^{54-r}</p><p><strong>Step 4:</strong> Simplifying: C(55,r)·2 = C(55,r+1)</p><p><strong>Step 5:</strong> Using C(n,r+1)/C(n,r) = (n-r)/(r+1):</p><p>2 = (55-r)/(r+1)</p><p>2(r+1) = 55-r</p><p>2r + 2 = 55 - r</p><p>3r = 53</p><p><strong>Step 6:</strong> Since r must be an integer and 53/3 ≈ 17.67, check r = 18: gives 3(18) = 54 ≠ 53. This suggests the problem statement may have n = 56 instead, giving r = 18.</p><p>∴ <strong>Answer: r = 18</strong> (assuming n = 56) <strong>or verify with corrected expansion parameters</strong></p>
Correct Answer: A