Sequences & Series for Matric Maths

Arithmetic, geometric, quadratic patterns and sigma notation for NSC Paper 1. CAPS-aligned with AI step-by-step explanations.

Sequences & Series in Paper 1 covers arithmetic and geometric sequences and series, quadratic (second-difference) patterns, sigma notation and infinite geometric series. Start every pattern question by checking first and second differences; that evidence tells you whether to use an arithmetic, geometric or quadratic rule before you substitute any values.

Key concepts

  • Arithmetic: Tn = a + (n − 1)d
  • Geometric: Tn = a·r^(n−1)
  • Sum formulas Sₙ for arithmetic and geometric series
  • Infinite geometric series: S∞ = a / (1 − r), |r| < 1
  • Quadratic patterns with constant second difference
  • Quadratic pattern Tn = an² + bn + c, where 2a equals the second difference
  • Tn = Sn − Sn−1
  • Convergence of a geometric series only when −1 < r < 1

Worked examples

For 2, 6, 18, 54, … find T₁₀.

Geometric with a = 2, r = 3. T₁₀ = 2·3⁹ = 39 366.

Find the general term of 3, 8, 15, 24, …

First differences: 5, 7, 9. Second difference: 2, so 2a = 2 and a = 1. 3a + b = 5 gives b = 2, and a + b + c = 3 gives c = 0. Tn = n² + 2n (check: T₄ = 16 + 8 = 24).

Calculate Σ (k = 1 to 20) of (3k + 2).

The terms are 5, 8, 11, …, 62: arithmetic with a = 5, d = 3 and n = 20. S₂₀ = 20/2 × (5 + 62) = 670.

For which values of x does Σ (n = 1 to ∞) of (2x − 1)ⁿ converge?

The ratio is r = 2x − 1. The series converges when −1 < 2x − 1 < 1, so 0 < 2x < 2 and 0 < x < 1.

An arithmetic sequence has T₃ = 11 and T₈ = 31. Find a and d.

T₈ − T₃ = 5d = 20, so d = 4. Then a + 2d = 11 gives a = 3.

Common mistakes

  • Forgetting the |r| < 1 condition for S∞.
  • Using arithmetic formulas on geometric patterns and vice-versa.
  • Miscounting terms in sigma notation: k = 3 to k = 10 is 8 terms, not 7.
  • Trying to write Tn for a quadratic pattern from the first differences only.

Exam tips

  • Check the first and second differences before deciding the pattern type.
  • For sigma notation, write out the first three terms to identify a and d or r.
  • Set up two equations in a and d (or a and r) whenever two terms are given.

In past papers

  • NSC Nov 2024 Paper 1 Q2 — sigma notation and infinite series

Frequently asked questions

What is sigma notation?

Σ is a compact way to write a sum. Σ from k=1 to n of aₖ means a₁ + a₂ + … + aₙ.

How do I know if a series converges?

A geometric series converges when −1 < r < 1, and its sum to infinity is S∞ = a/(1 − r). An arithmetic series with d ≠ 0 does not converge.

Are the sequence formulas on the information sheet?

Yes, the arithmetic and geometric Tn and Sn formulas and S∞ are given. The method for quadratic patterns is not, so learn how to find a, b and c.

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