Sequences & Series Formula Sheet for Grade 12 CAPS

Grade 12 sequences and series formulas for CAPS Maths: arithmetic and geometric sequences, sigma notation, sums and convergence for Matric Paper 1.

Number patterns are guaranteed marks in NSC Paper 1 because the formulas are supplied. The skill is choosing the correct one and identifying a, d or r correctly from the pattern.

Formulas

Formula	Expression	Note
Arithmetic general term	`Tₙ = a + (n − 1)d`	d = T₂ − T₁ must be constant.
Arithmetic series sum	`Sₙ = n/2 [2a + (n − 1)d]`	Use Sₙ = n/2 (a + l) when the last term is known.
Geometric general term	`Tₙ = a·rⁿ⁻¹`	r = T₂ ÷ T₁ must be constant.
Geometric series sum	`Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1`	Swap numerator and denominator signs when r < 1 to avoid negatives.
Sum to infinity	`S∞ = a/(1 − r), −1 < r < 1`	State the convergence condition to earn the reasoning mark.
Quadratic pattern general term	`Tₙ = an² + bn + c`	Second difference = 2a. Solve simultaneously for a, b and c.

How to use them

Always test whether the difference or the ratio is constant before choosing a formula.
For sigma notation, the number of terms is (upper − lower + 1).
Convergence questions almost always need −1 < r < 1 written explicitly.
Worked example: for 2 + 6 + 18 + …, a = 2 and r = 3, so S₆ = 2(3⁶ − 1)/(3 − 1) = 728.

Frequently asked questions

Which sequences and series formulas are used in Grade 12 Maths?

The key CAPS formulas cover arithmetic and geometric general terms, finite sums, sum to infinity and quadratic number patterns.

When can I use the sum to infinity formula?

Only for a geometric series where -1 < r < 1. State this condition in your working.

Are all Sequences & Series formulas given in the NSC exam?

Some are printed on the official information sheet, but not all. Treat every formula on this page as examinable and learn it with a worked example so you recognise when to use it.

Which NSC paper tests Sequences & Series?

Sequences & Series is examined in Paper 1 of the NSC Mathematics examination.

How do I practise these formulas?

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