IB Mathematics Analysis & Approaches HL

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant.

Common difference (d)

The constant difference between terms in an arithmetic sequence.

n^th term formula

The formula for finding the n^th term of an arithmetic sequence: u_n = u₁ + (n−1) × d.

Sum of an arithmetic sequence

The sum of the first n terms of an arithmetic sequence is given by S_n = (n / 2) × (u₁ + u_n).

Geometric sequence

A sequence in which the ratio between consecutive terms is constant.

Common ratio (r)

The constant ratio between terms in a geometric sequence.

n^th term formula for geometric sequence

The formula for finding the n^th term of a geometric sequence: u_n = u₁ × r^{(n - 1)}.

Sum of a geometric sequence

The sum of the first n terms of a geometric sequence is given by S_n = u₁ × (1 - rⁿ) / (1 - r), for r ≠ 1.

Infinite geometric series

The sum of an infinite geometric series when |r| < 1 is S_∞ = u₁ / (1 - r).

Sigma notation

A shorthand way of writing the sum of terms in a sequence using the symbol ∑.

Recurrence relation

Defines each term in a sequence using previous terms.

Product of powers

a^m × aⁿ = a^(m+n).

Quotient of powers

a^m / aⁿ = a^(m-n).

Power of a power

(a^m)ⁿ = a^(m×n).

Power of a product

(a × b)ⁿ = aⁿ × bⁿ.

Power of a quotient

(a / b)ⁿ = aⁿ / bⁿ.

Zero exponent

a⁰ = 1 (where a ≠ 0).

Negative exponent

a^(-n) = 1 / aⁿ.

Logarithm of a product

log_a(xy) = log_a(x) + log_a(y).

Logarithm of a quotient

log_a(x / y) = log_a(x) - log_a(y).

Change of base formula

log_a(x) = log_b(x) / log_b(a).

Binomial Theorem

The expansion of (a + b)ⁿ using binomial coefficients.

Pascal's Triangle

A triangular array of numbers that represents the binomial coefficients.

One-to-One Function

A function where each element in the domain maps to a distinct image in the co-domain.

Many-to-One Function

A function where multiple values in the domain map to the same element in the co-domain.

Onto Function

A function where every element in the co-domain has at least one pre-image in the domain.

Polynomial Function

A function that can be expressed in the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀.

Linear Function

A first-degree polynomial function of the form f(x) = mx + b.

Quadratic Function

A second-degree polynomial function of the form f(x) = ax² + bx + c.

Cubic Function

A third-degree polynomial function of the form f(x) = ax³ + bx² + cx + d.

Modulus Function

A function defined as f(x) = |x|, outputting the absolute value of x.

Identity Function

A function that maps every element to itself, f(x) = x.

Even Function

A function is even if f(x) = f(-x).

Odd Function

A function is odd if f(x) = -f(-x).

Composite Function

A function formed by combining two functions, denoted as f(g(x)).

Inverse Function

A function that 'undoes' the operation of the original function.

Critical points

Points where f'(x) = 0 or does not exist, used to find maximum or minimum values.

L'Hopital's Rule

A method to evaluate limits of indeterminate forms like 0/0 or ∞/∞.

Partial Fractions Decomposition

A method to break down a rational function into simpler fractions.

Discriminant

In the quadratic formula, it determines the nature of the roots: b² - 4ac.

Integrating Factor

A function used to simplify first-order linear differential equations.

Normal Distribution

A continuous probability distribution characterized by a bell-shaped curve.

Mean (μ)

The average of a set of values.

Variance (σ²)

The average of the squared differences from the mean.

Standard Deviation (σ)

The square root of the variance.

Pearson's Correlation Coefficient

A measure of the linear relationship between two variables.

Euler's Formula

e^(iθ) = cos(θ) + i sin(θ).

De Moivre's Theorem

Allows for the power of a complex number in polar form.