Sequences, series, and the sum to infinity (IB Math AI HL)

Key ideas

• Arithmetic: a fixed amount is added each time (uₙ = u₁ + (n − 1)d). Geometric: the amount is multiplied by r each time (uₙ = u₁rⁿ⁻¹).
• A decrease of 60% means r = 0.4, not 0.6: the multiplier is what is left.
• Sum to infinity u₁ ÷ (1 − r) is the long-term level, such as the amount of a drug in the body after many doses.
• For loans and savings with regular payments, use the finance solver (TVM) on the GDC; the series is what it calculates.
• To find when a total first passes a value, set up the inequality and use a table of values on the GDC.

Worked example

A patient takes a 20 mg dose of a drug every 12 hours. By the time of the next dose, 40% of the drug in the body remains. Find the amount in the body just after the 5th dose, and the long-term amount just after a dose.

1. Just after the nth dose, the amount is 20 + 20(0.4) + 20(0.4)² + … + 20(0.4)ⁿ⁻¹: a geometric series with u₁ = 20 and r = 0.4.
2. After 5 doses: S₅ = 20(1 − 0.4⁵) ÷ (1 − 0.4) = 20 × 0.98976 ÷ 0.6 ≈ 33.0 mg.
3. Long term: |r| < 1, so S∞ = 20 ÷ (1 − 0.4) = 33.3 mg (to 3 s.f.).
4. The amount gets very close to its long-term level within five doses.

Answer: About 33.0 mg after the 5th dose; the long-term amount is 100/3 ≈ 33.3 mg.

Exam-style HL AI questions on this topic

• Logs and Infinite Geometric Series questions — 51 questions with mark schemes, plus free worked examples
• All HL AI Number and Algebra questions by topic

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Sequences and sums on your calculator

Maths IA ideas that use sequences, series, and the sum to infinity

• How far does a bouncing ball really travel?
• Do TV series really get worse over time?
• Why are camera f-stops 1.4, 2, 2.8, 4, 5.6?
• Is deforestation in [a country] slowing down?
• All HL AI IA ideas

Related in Number and Algebra

• Exponential models for real-world decay
• Matrices — multiplication, inverse, and determinant
• Systems of linear equations and Gaussian elimination

Revise Sequences, series, and the sum to infinity for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S86 · Problem S93 · Problem S28 · Problem S92. All 22 →

Formulas in this topic: Sum to infinity of a geometric series. All formulas A–Z

Sibling topic in other IB Maths courses