Telescoping series

“Most of the middle collapses, so only the first and last survive.”

A series where consecutive terms cancel, leaving only the ends.

Memory trick: Split into differences.

Geometric mean

“Average growth rates multiply, so their fair average is geometric.”

The GM of a and b is √(ab).

√(ab)

Memory trick: √(ab), not (a+b)/2.

Circular permutations

“Seating friends around a round table fixes one and arranges the rest.”

n distinct objects arrange around a circle in (n−1)! ways.

(n−1)!

Memory trick: (n−1)! for a circle.

Combinations with a condition

“Sometimes it's easier to count what you don't want and subtract.”

Restricted selections use complementary counting: total minus the unwanted cases.

Memory trick: Count the complement.

Pascal's rule

“Pascal's triangle builds itself, row by row.”

nCr + nC(r−1) = (n+1)Cr — each triangle entry is the sum of the two above.

Memory trick: Add the two above.

Number of terms in an expansion

“One more term than the power — easy to remember.”

(a+b)^n has n+1 terms.

Memory trick: n+1 terms.

Locus

“A circle is the locus of points a fixed distance from a centre.”

The path traced by a point moving under a given condition.

Memory trick: Turn a condition into a curve.

Collinear points

“Zero enclosed area means they all lie on one line.”

Three points are collinear if the area of the triangle they form is zero.

Memory trick: Zero area → collinear.

General equation of a circle

“Read the centre and radius straight off the coefficients.”

x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).

Memory trick: Centre (−g,−f).

Condition of tangency

“Tangency is the knife-edge between missing and cutting through.”

A line touches a conic when it meets it in exactly one point (discriminant zero).

Memory trick: One intersection → tangent.

Asymptotes of a hyperbola

“The curve hugs its asymptotes ever more closely toward infinity.”

Straight lines the hyperbola approaches but never meets.

Memory trick: Approached, never touched.

Foci of an ellipse

“A gardener draws an ellipse with a loop of string around two pegs (foci).”

Two fixed points where the sum of distances to any point on the ellipse is constant.

Memory trick: Constant distance-sum.

Second derivative

“Acceleration is the second derivative of position.”

The derivative of the derivative; it measures how the slope itself changes (concavity).

d²y/dx²

Memory trick: Rate of change of the slope.

Tangent and normal

“A ball leaving a curved ramp flies off along the tangent.”

The tangent has slope f'(x); the normal is perpendicular to it.

Memory trick: Normal slope = −1/f'.

Approximation using differentials

“Engineers estimate small errors quickly using differentials.”

For a small change, Δy ≈ f'(x)·Δx.

Memory trick: Small change ≈ slope × step.

Logarithmic differentiation

“Logs turn a tangled product into an easy sum before differentiating.”

Take logs first to differentiate products, quotients or variable powers like x^x.

Memory trick: Log it, then differentiate.

Parametric differentiation

“It handles curves traced out over time, like a projectile's path.”

For x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt).

Memory trick: Divide the t-derivatives.

Definite integral as a limit of a sum

“Integration was born from adding up infinitely many slivers.”

The definite integral is the limit of Riemann sums of thin rectangles.

Memory trick: Sum of infinite strips.

Integral of even/odd functions

“Symmetry can kill an integral before you compute a thing.”

Over [−a,a], an odd function integrates to 0; an even one to twice the half-integral.

Memory trick: Odd over symmetric limits → 0.

Partial fractions

“Break one hard fraction into a few easy ones.”

Split a rational function into simpler fractions to integrate it.

Memory trick: Decompose, then integrate.

Linear differential equation

“The integrating factor is the magic multiplier that makes it solvable.”

dy/dx + Py = Q is solved using an integrating factor e^∫P dx.

Memory trick: Multiply by e^∫P dx.

Degree of a differential equation

“Clean it up first, then read off the degree.”

The power of the highest-order derivative after removing radicals and fractions.

Memory trick: Clear radicals first.

Coplanarity of vectors

“A zero box-volume means they lie flat in one plane.”

Three vectors are coplanar when their scalar triple product is zero.

Memory trick: [a b c] = 0 → coplanar.

Shortest distance between skew lines

“Two aircraft on skew paths miss by this closest gap.”

The length of the common perpendicular between two non-intersecting, non-parallel lines.

Memory trick: Along the common perpendicular.