"Least n such that…"
A standard method from Arithmetic and geometric sequences: the steps and worked examples.
Steps
- Write the sum or term as an inequality, for example Sₙ > 1000.
- AS: this is a quadratic inequality in n. Solve the quadratic, then take the next whole number.
- GS: rⁿ appears. Take logs: n log r > … (if 0 < r < 1, log r is negative, so the sign flips).
- Check the answer n and n − 1 in the original inequality.
Worked examples
Example 1: Same AS: find the least n such that Sₙ > 1000.
n/2 [10 + 3(n − 1)] > 1000
3n² + 7n − 2000 > 0
roots: n = [−7 ± √(49 + 24000)]/6, positive root ≈ 24.7
least n = 25 check: S₂₄ = 948, S₂₅ = 1025 ✓Example 2: GS: 3, 6, 12, … Find the least n such that Tₙ > 10 000.
3(2ⁿ⁻¹) > 10 000
2ⁿ⁻¹ > 3333.3…
(n − 1) log 2 > log 3333.3…
n − 1 > 11.70
least n = 13 check: T₁₂ = 6144, T₁₃ = 12 288 ✓For the whole topic, read Arithmetic and geometric sequences: key points, common mistakes and how the DSE tests it.