Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.

Step-by-step solution

Idea: a = 2 and r = 3. Set 2 × 3ⁿ⁻¹ = 4374 and write 4374 ÷ 2 as a power of 3.

1. a = 2, r = 62 = 3, so t_n = 2 × 3ⁿ⁻¹.½ mark
2. 2 × 3ⁿ⁻¹ = 4374 ⇒ 3ⁿ⁻¹ = 2187 = 3⁷.1 mark
3. n − 1 = 7 ⇒ n = 8. So 4374 is the 8th term.½ mark
4. Explicit formula: t_n = 2 × 3ⁿ⁻¹. Recursive formula: t₁ = 2, t_n = 3t_n−1 for n ≥ 2.1 mark

Check: 2, 6, 18, 54, 162, 486, 1458, 4374: the 8th term ✓.

Answer to write in the exam

Common mistakes that cost marks

• Concluding n = 7 from 3ⁿ⁻¹ = 3⁷. The exponent is n − 1, so n = 8.
• Dividing 4374 by 3 instead of by the first term 2 first.
• Writing the recursive rule as an AP rule (t_n = t_n−1 + 3).

How this can come in the exam

Try one yourself

Which term of the GP 5, 20, 80, … is 5120? Write the recursive formula.

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