Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.

Step-by-step solution

Idea: a = 2, r = 4. Set 2 × 4ⁿ⁻¹ = 131072 and write 65536 as a power of 4.

1. a = 2, r = 82 = 4, so t_n = 2 × 4ⁿ⁻¹.½ mark
2. 2 × 4ⁿ⁻¹ = 131072 ⇒ 4ⁿ⁻¹ = 65536 = 4⁸ (4⁴ = 256 and 256 × 256 = 65536).1 mark
3. n − 1 = 8 ⇒ n = 9: 131072 is the 9th term.½ mark
4. Explicit: t_n = 2 × 4ⁿ⁻¹; recursive: t₁ = 2, t_n = 4t_n−1 for n ≥ 2.1 mark

Check: 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072: the 9th term ✓.

Answer to write in the exam

Common mistakes that cost marks

• Stopping at 4ⁿ⁻¹ = 4⁸ and answering 8. The exponent is n − 1.
• Taking r = 6 (8 − 2) by subtracting instead of dividing.
• Writing 65536 = 4¹⁶ (confusing it with 2¹⁶). 2¹⁶ = (2²)⁸ = 4⁸.

How this can come in the exam

Try one yourself

Which term of the GP 3, 12, 48, … is 12288? Write the recursive formula.

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