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Sequences and series

Arithmetic and geometric progressions, nth terms, and sums of series.

  • Define and explain Sequences and series in your own words
  • Use key terms such as Arithmetic progression accurately
  • Apply what you have learned to new examples and questions
  • Avoid the common mistakes learners make with this topic

This lesson focuses on Sequences and series: arithmetic and geometric progressions, nth terms, and sums of series.

Definition: Sequences and series

Arithmetic and geometric progressions, nth terms, and sums of series.

Key ideas

Differentiating polynomials

From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule. So f(x) = 3x³ − 5x² + 2x − 7 gives f′(x) = 9x² − 10x + 2, and at x = 2 the gradient is 36 − 20 + 2 = 18.

Stationary points and nature

Solve f′(x) = 0 to find stationary points, then use the second derivative or a sign diagram: a positive second derivative means a minimum, a negative one a maximum. For f(x) = x³ − 3x, f′(x) = 3x² − 3 = 0 gives x = ±1, with a maximum at x = −1 and minimum at x = 1.

Key term — Arithmetic progression: A sequence where each term differs from the previous by a constant common difference, e.g. 3, 7, 11, 15.

Worked example: Sequences and series

Find the 10th term of the arithmetic sequence 5, 9, 13, 17, ….

5 + 9 × 4 = 5 + 36 = 41.

Answer: 5 + 9 × 4 = 5 + 36 = 41.

Common mistakes
  • Differentiating the constant term to 1 or x The derivative of any constant is 0 — it contributes no change.
  • Losing the constant of integration An indefinite integral must end with + C; different values of C give the whole family of antiderivatives.

Practice

Differentiate y = 4x³ + 6x.
Power rule on each term.

dy/dx = 12x² + 6.

Find ∫ (6x² + 4) dx.
Integrate term by term; don't forget + C.

2x³ + 4x + C.

Evaluate ∫₀³ 2x dx.
Integrate, then substitute the limits.

[x²]₀³ = 9 − 0 = 9.

Find the stationary points of y = x² − 6x + 5.
Set dy/dx = 0.

dy/dx = 2x − 6 = 0 → x = 3; stationary point (3, −4), a minimum since d²y/dx² = 2, which is positive.

Quick check

Sequences and series — quick check

Which of these best defines "Arithmetic progression"?

A sequence where each term differs from the previous by a constant common difference, e.g. 3, 7, 11, 15.

Find f(g(x)) given f(x) = x² and g(x) = 2x + 1.

f(g(x)) = (2x + 1)² = 4x² + 4x + 1.
Key takeaways
  • Sequences and series: arithmetic and geometric progressions, nth terms, and sums of series.
  • Differentiating polynomials: From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule.
  • Function: A rule that assigns each input exactly one output, written y = f(x); the set of allowed inputs is the domain.
  • Watch out for: differentiating the constant term to 1 or x
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