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Mathematics · 7. Sequences and series

Arithmetic and geometric progressions and growth models

Recognise patterns, build nth-term rules, calculate finite sums and test convergence of infinite geometric series.

45 minutesContent version: 2.2
Start the interactive pathwayOpen the virtual experiment

Driving question

How can we distinguish constant addition from proportional growth and choose the correct model?

Curriculum coverage

Vietnam Mathematics 11; Cambridge IGCSE 0580 Extended and AS & A Level 9709: sequences, series and growth models.

Model scope and limitations

Progression formulas apply only when the difference or ratio is genuinely constant. Compound-growth models assume a fixed rate and compounding interval; real data may violate those assumptions.

Learning objectives

  • Derive nth terms for arithmetic and geometric progressions.
  • Calculate finite sums and solve for n under a real constraint.
  • Use an infinite sum only when |q| < 1 and interpret its limit.

Prerequisite knowledge

  • Powers, equations and algebraic manipulation.
  • Percentages and basic logarithms when solving for the number of periods.

Core knowledge

Arithmetic progression

If uₙ₊₁ − uₙ = d is constant, then uₙ = u₁ + (n−1)d and Sₙ = n[2u₁+(n−1)d]/2.

Geometric progression

If uₙ₊₁/uₙ = q is constant, then uₙ = u₁qⁿ⁻¹ and Sₙ = u₁(1−qⁿ)/(1−q) for q ≠ 1.

Infinite geometric sum

When |q| < 1, qⁿ → 0 and S∞ = u₁/(1−q). If |q| ≥ 1 and u₁ ≠ 0, partial sums do not approach a finite value.

Worked example

A geometric progression has u₁ = 3 and u₄ = 24. From 3q³ = 24, q³ = 8 and q = 2. The first six terms sum to S₆ = 3(2⁶−1)/(2−1) = 189. The infinite-sum formula is invalid because |q| = 2.

Misconceptions to avoid

  • The nth arithmetic term contains (n−1)d, not nd, because no increase has occurred at the first term.
  • A 5% increase per period multiplies by 1.05; it is not a fixed addition equal to 5% of the initial value.
  • Having uₙ → 0 is not sufficient for ∑uₙ to converge; it is only a necessary condition.

Virtual experiment procedure: Sequences and series explorer

  1. Enter at least five terms and inspect consecutive differences and ratios before choosing a model.
  2. Predict uₙ and Sₙ algebraically, then compare with the simulation table or graph.
  3. Test q on both sides of |q| = 1 to observe convergent or divergent partial sums.

Safety and cautions

  • A finite plot may appear stable even when a sequence does not converge; always check the definition and algebraic condition.

Evidence to collect

  • A table of n, uₙ and Sₙ for at least five values, with the generating formulas.
  • A difference/ratio check and a conclusion stating the convergence condition when an infinite sum is used.
Open the virtual experiment
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<iframe src="https://visuallab.win/math-lab/?module=sequences_series" width="100%" height="560" frameborder="0" allow="fullscreen" loading="lazy"></iframe>

Diagnostic check

1. What type is 4, 7, 10, 13, …?

  1. Arithmetic with d = 3
  2. Geometric with q = 3
  3. No pattern
Show answer and explanation

Correct answer: Arithmetic with d = 3

Every consecutive difference is 3.

2. What is the convergence condition for an infinite geometric series?

  1. |q| < 1
  2. q > 1
  3. q ≠ 0
Show answer and explanation

Correct answer: |q| < 1

Only |q| < 1 makes qⁿ tend to zero so the partial sums have a finite limit.

3. A quantity increasing by 8% per year has what common ratio?

  1. 1.08
  2. 0.08
  3. 8
Show answer and explanation

Correct answer: 1.08

The new value is 100% + 8% = 108% of the old value.

Application practice

1. An arithmetic progression has u₁ = 5 and d = 3. What is u₁₀?

  1. 32
  2. 35
  3. 27
Show answer and explanation

Correct answer: 32

u₁₀ = 5 + (10−1)×3 = 32.

2. A geometric progression has u₁ = 2 and q = 3. What is S₄?

  1. 80
  2. 54
  3. 40
Show answer and explanation

Correct answer: 80

S₄ = 2(3⁴−1)/(3−1) = 80.

3. A ball rebounds to 60% of its previous height from an initial height of 10 m. What is its ideal total distance over infinitely many bounces?

  1. 40 m
  2. 25 m
  3. 16 m
Show answer and explanation

Correct answer: 40 m

The initial fall is 10 m; later up-and-down distances total 2(6+3.6+…) = 2×6/(1−0.6)=30 m, giving 40 m.

Academic references

  1. Cambridge IGCSE Mathematics 0580 — official syllabus and resources
  2. Cambridge AS & A Level Mathematics 9709 — official syllabus
  3. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

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WebLab STEM
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