Arithmetic Sequences & Series
Master the logic of constant growth — from general term to sum formulas with deep explanations and fully worked examples.
Sequences and Series
Table of Contents
To master sequences and series, you must understand the core logic of how patterns grow. This isn’t just about memorizing formulas; it’s about seeing the “DNA” of a number pattern.
Why this matters#
Sequences and series are the language of growth and prediction. Whether it’s the interest in a bank account (Geometric), the steady steps of a staircase (Arithmetic), or the path of a projectile (Quadratic), these patterns allow us to map reality onto math.
The Big Picture#
- Arithmetic: Linear growth. Steady, predictable, and unchanging speed.
- Geometric: Exponential growth or decay. Scaling, doubling, or halving. Can have alternating signs.
- Quadratic: Acceleration. Patterns where the gaps themselves are changing. Think of a parabola that can also have negative and positive values on a dip.
- Sigma: The tool for summing it all up.
Navigation#
Below, you’ll find deep dives into each pattern type. We recommend starting with Arithmetic to understand the basics of growth, before moving to the scaling logic of Geometric and the acceleration of Quadratic.
- Arithmetic Sequences & Series: Constant first differences ($ T_n = a + (n-1)d $).
- Quadratic Sequences: Constant second differences ($ T_n = an^2 + bn + c $).
- Geometric Sequences & Series: Constant ratios ($ S_\infty $).
- Sigma Notation: Understanding $ \Sigma $ sums.
- Mixed & Combined Sequences: Handling alternating patterns.
⚠️ You MUST Know the Proofs#
The CAPS curriculum requires you to prove the sum formulas. These proofs are regularly examined and are worth easy marks if you’ve practised them. Do NOT skip them.
| Proof | What you must show |
|---|---|
| Arithmetic Series ($S_n$) | Write the series forwards and backwards, add them, and show $S_n = \frac{n}{2}(2a + (n-1)d)$ |
| Geometric Series ($S_n$) | Multiply the series by $r$, subtract from the original, and factorise to get $S_n = \frac{a(r^n - 1)}{r - 1}$ |
| Sum to Infinity ($S_\infty$) | Start from $S_n = \frac{a(1 - r^n)}{1 - r}$, show that as $n \to \infty$ and $\lvert r \rvert < 1$, $r^n \to 0$, giving $S_\infty = \frac{a}{1-r}$ |
Exam tip: Familiarise yourself with every step of the proofs — marks are awarded for each line, not just the final answer.
Build on Your Lower-Grade Foundations#
If number patterns still feel shaky, revise these lower-grade pages first:
- Grade 10 Number Patterns — linear patterns and the general term
- Grade 10 Fundamentals: Basic Algebra
- Grade 11 Number Patterns — quadratic patterns and second differences
- Grade 11 Fundamentals: Equation Solving
📚 Quick Study Tips#
- Quadratic sequences: Always remember the $ 2a $, $ 3a+b $, $ a+b+c $ method.
- Geometric Series: Check for convergence ($ -1 < r < 1 $) before using $ S_\infty $.
- Sigma: Always verify the number of terms ($ \text{Top} - \text{Bottom} + 1 $).
📄 Past Papers#
Practice is key! You can find official Grade 12 Past Exam Papers on the Department of Basic Education website. Look for Mathematics Paper 1 for Questions on Sequences and Series.
⏮️ Algebra | 🏠 Back to Grade 12 | ⏭️ Functions & Inverses
Quadratic Sequences
Master the logic of changing speed — first and second differences, the general term derivation, and solving for n with fully worked examples.
Geometric Sequences & Series
Master the logic of scaling — common ratio, general term, sum formulas with proofs, sum to infinity, and convergence with deep explanations and fully worked examples.
Sigma Notation
Master sigma notation from the ground up — decoding the symbol, expanding, counting terms, connecting to sum formulas, and splitting sums with fully worked examples.
Mixed & Combined Sequences
Master alternating and interleaved patterns — how to untangle two sequences woven together and find any term or sum.