Find a and d (or a and r)
A standard method from Arithmetic and geometric sequences: the steps and worked examples.
Steps
- Turn each given fact into an equation using the formula card.
- Two facts give two equations. Solve simultaneously.
- For a GS, divide one equation by the other to cancel a.
- Substitute back to find the other unknown, then answer the question (Tₙ or Sₙ).
Worked examples
Example 1: AS: a = 5, d = 3. Find T₂₀ and S₂₀.
T₂₀ = 5 + 19(3) = 62
S₂₀ = 20/2 (5 + 62) = 670Example 2: 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 3: GS: 12 + 6 + 3 + … Find the sum to infinity.
a = 12, r = 1/2, so S∞ = 12/(1 − ½) = 24.
Example 4: 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 ✓Example 5: Find a and d: an AS has T₃ = 11 and T₈ = 26. Find a, d and T₂₀. Then, for a GS with T₂ = 6 and T₅ = 48, find a and r.
AS: a + 2d = 11
a + 7d = 26
subtract: 5d = 15, d = 3; a = 11 − 6 = 5
T₂₀ = 5 + 19 × 3 = 62
GS: ar = 6
ar⁴ = 48
divide: r³ = 8, r = 2; a = 6 ÷ 2 = 3For the whole topic, read Arithmetic and geometric sequences: key points, common mistakes and how the DSE tests it.