Sum of infinite GP

“Add ½ + ¼ + ⅛ + … forever and you get exactly 1.”

Converges to a/(1−r) only when |r| < 1.

S∞=a/(1−r)

Memory trick: Needs |r| < 1.

AM ≥ GM inequality

“A powerful shortcut for finding minimum or maximum values.”

The arithmetic mean of positive numbers is at least their geometric mean.

Memory trick: AM ≥ GM, equal only when all equal.

Fundamental counting principle

“3 shirts and 4 trousers make 12 outfits — multiplication, not addition.”

If one task has m ways and another n ways, together they have m×n ways.

Memory trick: 'And' multiplies; 'or' adds.

Permutation

“The number of ways to award gold, silver and bronze — order matters.”

An ordered arrangement of r from n: nPr = n!/(n−r)!.

nPr=n!/(n−r)!

Memory trick: Order matters → permutation.

Combination

“Choosing a committee — the order you pick people doesn't matter.”

An unordered selection of r from n: nCr = n!/[r!(n−r)!].

nCr=n!/(r!(n−r)!)

Memory trick: Order irrelevant → combination.

Binomial theorem

“Pascal's triangle hands you the coefficients row by row.”

(a+b)^n expands with coefficients nCr, powers of a and b summing to n.

Tᵣ₊₁=nCr·a^(n−r)b^r

Memory trick: Exponents add to n each term.

General term of a binomial

“To find a specific term, set the exponent and solve for r.”

The (r+1)th term is nCr·a^(n−r)·b^r.

Tᵣ₊₁

Memory trick: Mind the +1 in the index.

Pythagorean identity

“The unit circle in one equation.”

For any angle, sin²θ + cos²θ = 1.

sin²θ+cos²θ=1

Memory trick: It's the squares that sum to 1.

Radian measure

“Radians tie an angle to the arc length it sweeps, which is why calculus loves them.”

π radians = 180°; calculus formulas assume radians.

π rad = 180°

Memory trick: Switch to radians before differentiating.

Range of sine and cosine

“If your working gives sinθ = 1.4, you've slipped somewhere.”

sinθ and cosθ always lie between −1 and 1.

Memory trick: Never outside [−1, 1].

Sine and cosine addition

“These unlock angles you can't read straight off the triangle.”

sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB.

Memory trick: Watch the sign flip in cosine.

Sine rule

“It cracks triangles when you know an angle and its opposite side.”

In any triangle, a/sinA = b/sinB = c/sinC.

a/sinA=b/sinB

Memory trick: Pair each side with its opposite angle.

Cosine rule

“When there's no right angle, the cosine rule steps in.”

c² = a² + b² − 2ab·cosC generalises Pythagoras to any triangle.

c²=a²+b²−2ab·cosC

Memory trick: Pythagoras with a correction term.

Distance formula

“It's Pythagoras wearing coordinate-geometry clothes.”

The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²].

d=√(Δx²+Δy²)

Memory trick: Difference-of-coordinates, squared, rooted.

Slope of a line

“Every rate you plot — speed, cost, growth — is a slope.”

Slope m = (y₂−y₁)/(x₂−x₁) = tanθ measures steepness.

m=Δy/Δx

Memory trick: Rise over run.

Parallel and perpendicular lines

“Perpendicular slopes are negative reciprocals of each other.”

Parallel lines share a slope; perpendicular slopes multiply to −1.

Memory trick: ⊥ → m₁m₂ = −1.

Equation of a circle

“Every point equidistant from the centre — written as an equation.”

A circle of centre (h,k) and radius r is (x−h)² + (y−k)² = r².

(x−h)²+(y−k)²=r²

Memory trick: Centre-radius form.

Parabola

“Satellite dishes are parabolas because they focus all signals to one point.”

The set of points equidistant from a focus and a directrix; y² = 4ax opens rightward.

y²=4ax

Memory trick: Eccentricity exactly 1.

Eccentricity of conics

“A single number e morphs a circle into an ellipse, parabola, then hyperbola.”

Circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1.

Memory trick: e climbs, the curve opens up.

Limit

“Mathematics' way of getting infinitely close without touching.”

The value a function approaches as its input nears a point, even if undefined there.

Memory trick: Limit ≠ value at a discontinuity.

Continuity

“You can draw a continuous curve without lifting your pen.”

A function is continuous at a point if its limit there equals its value — no breaks.

Memory trick: Limit = value → continuous.

Standard limit sinx/x

“The tiny-angle secret behind much of calculus and physics.”

lim(x→0) sin x / x = 1 (x in radians).

lim(x→0)sinx/x=1

Memory trick: Only in radians.

Derivative

“Your speedometer shows a live derivative of distance.”

The instantaneous rate of change — the slope of the tangent at a point.

dy/dx

Memory trick: Slope of the tangent.

Power rule

“The workhorse of differentiation — bring the power down, drop it by one.”

d/dx(x^n) = n·x^(n−1).

d/dx(xⁿ)=nx^(n−1)

Memory trick: Drop the power, subtract one.