Find  the $n^{th}$ term of the sequence 2 x 3, 4 x 6, 8 x 9, 16 x 12...

Analyze the question: Each term is a product of two numbers. We find the pattern in each part separately.

Step 1: First factors 2, 4, 8, 16, ... is a geometric progression with first term 2 and common ratio 2. The nth term is 2ⁿ.

Step 2: Second factors 3, 6, 9, 12, ... is an arithmetic progression with first term 3 and common difference 3. The nth term is 3 + (n − 1)3 = 3n.

Step 3: Combine The nth term is 2ⁿ × 3n.

Check:

• n = 1: 2¹ × 3(1) = 2 × 3 ✓
• n = 2: 2² × 3(2) = 4 × 6 ✓
• n = 3: 2³ × 3(3) = 8 × 9 ✓
• n = 4: 2⁴ × 3(4) = 16 × 12 ✓

Why the others fail:

• A: 2ⁿ × 3(n + 1) gives 2 × 6 for n = 1, which is wrong.
• C: 2ⁿ × 3ⁿ gives 2 × 3, then 4 × 9, which is wrong.
• D: 2ⁿ × 3ⁿ⁻¹ gives 2 × 1 for n = 1, which is wrong.

Tip: When a sequence is a product of two patterns, find the rule for each part separately (GP or AP), then multiply. Always test n = 1 and n = 2 in the options to eliminate wrong ones quickly.