1 · IB Maths
Sequences and series
Arithmetic sequences add a common difference; geometric ones multiply by a common ratio. Watch the terms and their running sum build, and see when a geometric series settles to a finite sum.
Predict first
Choose the preset 'Converging (r = 0.6)' (u₁ = 5). You increase the common ratio r from 0.6 to 0.8. What happens to the sum to infinity S∞?
S∞ = u₁ ÷ (1 − r). With r = 0.6 this is 5 ÷ 0.4 = 12.5; with r = 0.8 it is 5 ÷ 0.2 = 25, so a small change in r near 1 has a large effect.
Check it in the model: Pick the preset, then move the Common ratio, r slider from 0.6 to 0.8 and read the Sum to infinity S∞ line.
terms uₖ running sum Sₖ (own scale) sum to infinity
Sₙ = (n ÷ 2)(2u₁ + (n − 1)d)
Geometric: uₙ = u₁ rⁿ⁻¹
Sₙ = u₁(rⁿ − 1) ÷ (r − 1), r ≠ 1
S∞ = u₁ ÷ (1 − r), only if |r| < 1
Try this
- Choose 'Arithmetic +6' and change n from 8 to 12. Check that S₁₂ = 6 × (2 × 2 + 11 × 6) = 420 against the formula.
- Choose 'Converging (r = 0.6)' with the running sum on, and increase n. How close does Sₙ get to S∞ after 12 terms?
- Set the type to geometric and move r through −1, 0 and 1 (with a = 2). What happens to the terms and to S∞ at each value?
Common error: the sum to infinity u₁ ÷ (1 − r) exists only when |r| < 1, so state this condition before using the formula, and remember that r = 1 needs Sₙ = n u₁ instead.
More on 1: Arithmetic and geometric series · Compound interest and depreciation · Proof by induction · Permutations and combinations · De Moivre's theorem · Systems of linear equations · Complex numbers on the Argand plane · Roots of a complex number · Complex loci · Matrix transformations · Exponentials and logarithms · Binomial theorem