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Grade 11 — Formula Reference Sheet

Grade 11 — CAPS · 17 formulas

Grade 11 — CAPS

Grade 11

Sequences & Series, Compound Finance, Trigonometric identities, Probability, Euclidean Geometry.

17 formulas · opens your print dialog

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Sequences & Series

Arithmetic nth Term

T_{n} = a + (n - 1) d

Finds the nth term of an arithmetic (linear) sequence.

When to use

Any pattern where the difference between consecutive terms is constant.

Example

Sequence 3, 7, 11, … (a=3, d=4): T₁₀ = 3 + 9×4 = 39

Sequences & Series

Arithmetic Series Sum

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

Sum of the first n terms of an arithmetic series.

When to use

Adding up a set of numbers that form an arithmetic pattern.

Example

Sum of 3+7+11+… (10 terms): S₁₀ = 5×(6 + 36) = 210

Sequences & Series

Geometric nth Term

T_{n} = a r^{n - 1}

Finds the nth term of a geometric (exponential) sequence.

When to use

Any pattern where each term is multiplied by a constant ratio r.

Example

Sequence 2, 6, 18, … (a=2, r=3): T₅ = 2×3⁴ = 162

Sequences & Series

Geometric Series Sum

S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1} for r \neq = 1

Sum of the first n terms of a geometric series.

When to use

Adding up terms of a geometric sequence.

Example

Sum of 2+6+18+… (5 terms): S₅ = 2(3⁵−1)/(3−1) = 242

Financial Mathematics

Compound Interest (Growth)

A = P (1 + i)^{n}

Amount after compound interest — interest is added to the principal each period.

When to use

Savings accounts, investments, any scenario where interest compounds.

Example

R10 000 at 7% p.a. for 5 years: A = 10000(1.07)⁵ = R14 025.52

Financial Mathematics

Compound Decay (Depreciation)

A = P (1 - i)^{n}

Value after compound depreciation — asset loses a percentage of its current value each period.

When to use

Reducing-balance depreciation of vehicles and equipment.

Example

R200 000 at 20% for 4 years: A = 200000(0.8)⁴ = R81 920

Financial Mathematics

Effective Annual Interest Rate

(1 + i_{e} f f) = (1 + \frac{i _{n o m}}{m})^{m}

Converts a nominal rate compounded m times per year to an annual effective rate.

When to use

Comparing interest rates that compound at different frequencies.

Example

12% p.a. compounded monthly: iₑff = (1 + 0.01)¹² − 1 = 12.68%

Trigonometry

Sine Rule

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

Relates sides of any triangle to the sines of their opposite angles.

When to use

Given two angles and a side, or two sides and a non-included angle.

Example

If A=30°, B=45°, a=8: b = 8×sin45°/sin30° = 11.31

Trigonometry

Cosine Rule

a^{2} = b^{2} + c^{2} - 2 b c \cdot cos A

Finds an unknown side or angle in any triangle given enough information.

When to use

Given two sides and the included angle, or three sides.

Example

b=5, c=7, A=60°: a² = 25+49−70cos60° = 39, a = 6.24

Trigonometry

Area of a Triangle (Trig)

A r e a = \frac{1}{2} ab \cdot sin C

Finds the area when two sides and the included angle are known.

When to use

Area problems where the height is not given directly.

Example

a=6, b=8, C=30°: Area = ½×6×8×0.5 = 12 square units

Trigonometry

Double Angle — Sine

sin 2 α = 2 sin α cos α

Expresses the sine of a double angle in terms of sine and cosine of the original angle.

When to use

Simplifying expressions, proving identities involving sin 2α or 2sinα cosα.

Example

sin 60° = 2 sin 30° cos 30° = 2×0.5×(√3/2) = √3/2 ✓

Trigonometry

Double Angle — Cosine

cos 2 α = cos^{2} α - sin^{2} α = 1 - 2 s i n^{2} α = 2 co s^{2} α - 1

Three equivalent forms for the cosine of a double angle.

When to use

Proving identities, solving equations involving cos 2α.

Example

cos 90° = 1 − 2sin²45° = 1 − 2×½ = 0 ✓

Probability

Addition Rule

P (A or B) = P (A) + P (B) - P (A and B)

Probability that event A or B occurs (or both).

When to use

When events are not mutually exclusive (they can both happen).

Example

P(A)=0.4, P(B)=0.5, P(A∩B)=0.2: P(A∪B) = 0.7

Probability

Complement Rule

P (A^{'}) = 1 - P (A)

Probability of an event NOT occurring.

When to use

When it is easier to calculate the probability of the opposite event.

Example

P(rolling a 6) = 1/6, so P(not 6) = 5/6

Probability

Conditional Probability

P (A ∣ B) = \frac{P ( A and B )}{P ( B )}

Probability of A given that B has already occurred.

When to use

When knowledge of one event affects the probability of another.

Example

P(rain|cloudy) = P(rain and cloudy) / P(cloudy)

Analytical Geometry

Equation of a Circle

(x - a)^{2} + (y - b)^{2} = r^{2}

Standard form of a circle with centre (a, b) and radius r.

When to use

Describing circles on the Cartesian plane, finding centres and radii.

Example

Centre (2, −3), radius 5: (x−2)² + (y+3)² = 25

Statistics

Standard Deviation

σ = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}

Measures how spread out data is around the mean.

When to use

Comparing variability between data sets; quality control problems.

Example

Data {2, 4, 4, 4, 5, 7}: x̄=4.33, σ≈1.49

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