What is covered in Grade 12 Mathematics Sequences & Series?

Arithmetic and geometric sequences: general term and sum. Sigma notation. Infinite geometric series and convergence.

Is Grade 12 Sequences & Series tested in Paper 1 or Paper 2?

Grade 12 Mathematics Sequences & Series is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Sequences & Series taught in Grade 12 Mathematics?

Sequences & Series is taught in Term 1 of Grade 12 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 12 Mathematics Sequences & Series on the CAPS curriculum?

Yes. Sequences & Series is part of the official CAPS Mathematics curriculum for Grade 12, Term 1 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 12 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 12 Mathematics

Grade 12 · Term 1Mathematics

Sequences & Series

We derive and apply formulae for arithmetic and geometric series (sums), investigate convergence of infinite geometric series, and apply sigma notation.

Week 1

1.1 Arithmetic & Geometric Series

Derive and apply the sum of an arithmetic series: $S_n=\frac{n}{2}(2a+(n-1)d)$
Derive and apply the sum of a geometric series: $S_n=\frac{a(r^n-1)}{r-1}$
Apply sigma notation $\sum$

🌍

Real-World Connection

The geometric series formula is behind every mortgage calculation. Each monthly payment partially covers interest (growing exponentially) and partially repays principal — the formula tells the bank the exact balance at any point, producing the amortisation schedule you see on your bond statement.

Arithmetic Series Sum

S_n = \frac{n}{2}(2a+(n-1)d) = \frac{n}{2}(a+l)

$a$ = first term, $d$ = common difference, $l=T_n$ = last term, $n$ = number of terms

Geometric Series Sum

S_n = \frac{a(r^n-1)}{r-1} = \frac{a(1-r^n)}{1-r}

$a$ = first term, $r$ = common ratio, $r\neq1$

Infinite Geometric Series

S_\infty = \frac{a}{1-r} \quad \text{only if } |r| < 1

Converges only when $-1<r<1$; diverges otherwise

Definition

Sigma Notation

The symbol $\sum$ (sigma) denotes a sum. The index variable runs from the lower limit to the upper limit.

\sum_{k=1}^{n} T_k = T_1+T_2+\ldots+T_n

💡 Tip

When $r=1$ in the geometric series formula, the denominator is 0. But when $r=1$ , every term equals $a$ , so $S_n=na$ . Use this special case directly.

Worked Examples

Worked Example

Arithmetic series sum

Problem

Find the sum of the arithmetic series: 3+7+11+\ldots+99

Worked Example

Infinite geometric series

Problem

S_\infty

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge3 marks

Find

\sum_{k=1}^{5}(2k+1)

L1 · Knowledge1 mark

State the condition for a geometric series to converge.

L2 · Routine Procedures3 marks

Find

S_{10}

for the arithmetic series

4+7+10+\ldots

L2 · Routine Procedures3 marks

Find

S_6

for the geometric series

2+6+18+\ldots

L3 · Complex Procedures4 marks

The sum of the first

n

terms of an arithmetic series is

S_n=3n^2+2n

. Find

T_{10}

L3 · Complex Procedures4 marks

Find the sum to infinity:

\sum_{n=1}^{\infty}3\cdot\left(-\frac{2}{5}\right)^{n-1}

L4 · Problem Solving5 marks

An arithmetic series has first term

a

and common difference

d

. If

S_5=60

and

S_{10}=195

, find

a

and

d

L4 · Problem Solving4 marks

Prove that

\displaystyle\sum_{k=1}^{n}k = \dfrac{n(n+1)}{2}

using an arithmetic series argument.

Functions & Inverses