<p>Consider a sequence \(\{a_n\}\) with \(a_1 = 2\) and \(a_n = \dfrac{a_{n-1}^2}{a_{n-2}}\) for all \(n \geq 3\), terms of the sequence being distinct. Given that \(a_2\) and \(a_5\) are positive integers and \(a_5 \leq 162\), then the possible value(s) of \(a_5\) can be</p>
Step-by-Step Solution
Key Concept: Express each term using the recurrence relation to find a pattern in terms of a₁ and a₂, then use divisibility constraints and the bound a₅ ≤ 162 to determine which integer values work.
<p><strong>Step 1:</strong> Let a₁ = 2 and a₂ = k (positive integer). Use the recurrence aₙ = a²ₙ₋₁/aₙ₋₂.</p><p><strong>Step 2:</strong> Calculate successive terms:</p><p>• a₃ = a₂²/a₁ = k²/2</p><p>• a₄ = a₃²/a₂ = (k⁴/4)/k = k³/4</p><p>• a₅ = a₄²/a₃ = (k⁶/16)/(k²/2) = k⁴/8</p><p><strong>Step 3:</strong> For a₃ and a₅ to be positive integers:</p><p>• k²/2 must be an integer ⟹ k is even, say k = 2m</p><p>• Then a₅ = (2m)⁴/8 = 16m⁴/8 = 2m⁴</p><p><strong>Step 4:</strong> Apply constraint a₅ ≤ 162:</p><p>2m⁴ ≤ 162 ⟹ m⁴ ≤ 81 ⟹ m ≤ 3</p><p><strong>Step 5:</strong> Check distinctness condition for m ∈ {1, 2, 3}:</p><p>• m = 1: a₁ = 2, a₂ = 2 → NOT distinct ✗</p><p>• m = 2: a₁ = 2, a₂ = 4, a₃ = 8, a₄ = 16, a₅ = 32 → All distinct ✓</p><p>• m = 3: a₁ = 2, a₂ = 6, a₃ = 18, a₄ = 108, a₅ = 162 → All distinct ✓</p><p><strong>Step 6:</strong> Verify a₅ values:</p><p>• m = 2: a₅ = 2(2)⁴ = 32</p><p>• m = 3: a₅ = 2(3)⁴ = 162</p><p>∴ Answer: A, B (corresponding to a₅ = 32 and a₅ = 162)</p>
Correct Answer: A,B