<p>Find the number of common terms in the following sequences: (i) 3, 7, 11, … to 100 terms and 2, 5, 8, … to 100 terms.</p>
Step-by-Step Solution
Key Concept: Common terms form an arithmetic progression themselves. Find the general terms of both sequences, set them equal to find the pattern of common terms, then count how many satisfy both range constraints.
<p><strong>Step 1:</strong> Write general terms for both sequences.</p><p>Sequence 1: a₁ = 3, d₁ = 4 → T₁(m) = 3 + 4(m-1) = 4m - 1, where m = 1 to 100</p><p>Sequence 2: a₂ = 2, d₂ = 3 → T₂(n) = 2 + 3(n-1) = 3n - 1, where n = 1 to 100</p><p><strong>Step 2:</strong> Set them equal to find common terms.</p><p>4m - 1 = 3n - 1 → 4m = 3n → n = 4m/3</p><p>For n to be an integer, m must be divisible by 3. Let m = 3k for k ≥ 1.</p><p>Then n = 4k and common term = 4(3k) - 1 = 12k - 1</p><p><strong>Step 3:</strong> Apply constraints from both sequences.</p><p>From Sequence 1: m = 3k ≤ 100 → k ≤ 33.33 → k ≤ 33</p><p>From Sequence 2: n = 4k ≤ 100 → k ≤ 25</p><p><strong>Step 4:</strong> Find valid range for k.</p><p>We need k ≥ 1 and k ≤ 25 (the stricter constraint)</p><p>Common terms: 11, 23, 35, ..., 299 (when k = 1, 2, 3, ..., 25)</p><p>∴ Number of common terms = <strong>25</strong></p>
Correct Answer: 25