Binomial Theorem & Series cheatsheet (IB Maths AA HL)

The Binomial Theorem is one of the most reliable mark-earners in IB Mathematics Analysis & Approaches HL — but only if you know where the traps are. The SL version (positive integer $n$, finite expansion) is heavily tested in Paper 1 and almost always asks for a specific term using the general term $T_{r+1}$. The AHL extension (rational or negative $n$) introduces the generalised binomial series, which links directly to the Maclaurin series, partial fractions, and the geometric series — and brings the brutal validity condition $|x| < 1$ that costs students a guaranteed mark when forgotten.

This cheatsheet condenses every formula, identity, technique, trick and trap from SL 1.3 and AHL 1.9 into one revision page.

§1 — Combinations & the Binomial Theorem SL 1.3

The binomial coefficient

$$\binom{n}{r} = C^n_r = \dfrac{n!}{r!\,(n-r)!}$$

• $\binom{n}{0} = \binom{n}{n} = 1$
• $\binom{n}{r} = \binom{n}{n-r}$   (symmetry)
• Pascal: $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$

P1 step-back method (no calculator)

Rule: Start at $n$, multiply $r$ factors downward, divide by $r!$.

Examples: $\binom{10}{2} = \dfrac{10 \times 9}{2} = 45$   ·   $\binom{8}{3} = \dfrac{8 \times 7 \times 6}{6} = 56$.

Pascal's triangle

Row $n$ gives the coefficients of $(a + b)^n$. Sum of row $n$ $= 2^n$.

Key identities

Sum of even-indexed $=$ sum of odd-indexed $= 2^{n-1}$.

§2 — The Binomial Theorem & General Term SL 1.3

$$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r, \qquad n \in \mathbb{N}$$

General term (most important formula):

$$\boxed{\;T_{r+1} = \binom{n}{r}\, a^{n-r}\, b^r\;}$$

Properties of the expansion:

• Exactly $n + 1$ terms.
• Powers of $a$ decrease $n \to 0$.
• Powers of $b$ increase $0 \to n$.
• Every term: $(n - r) + r = n$. ✓
• Coefficients are symmetric.

5-step attack: finding a specific term

1. Write $T_{r+1} = \binom{n}{r}\, a^{n-r}\, b^r$ with your $a$, $b$, $n$ substituted.
2. Collect the power of $x$ — write it as a function of $r$.
3. Set that power equal to what you need (e.g. 3 for $x^3$, 0 for the constant term).
4. Solve for $r$ — must be a non-negative integer $\leq n$.
5. Substitute back and compute (including $\binom{n}{r}$, numerical factors, sign).

Worked micro-examples

Find the term in $x^3$ in $(2x - 1)^7$: $a = 2x$, $b = -1$, $n = 7$.
$T_{r+1} = \binom{7}{r}(2x)^{7-r}(-1)^r = \binom{7}{r}\, 2^{7-r}(-1)^r\, x^{7-r}$.
Set $7 - r = 3 \Rightarrow r = 4$: $T_5 = \binom{7}{4} \cdot 2^3 \cdot (-1)^4 \cdot x^3 = 35 \cdot 8 \cdot 1 \cdot x^3 = 280 x^3$.

Constant term in $\left(x^2 - \dfrac{2}{x}\right)^6$:
$T_{r+1} = \binom{6}{r}(x^2)^{6-r}(-2/x)^r = \binom{6}{r}(-2)^r x^{12 - 3r}$.
Set $12 - 3r = 0 \Rightarrow r = 4$: $T_5 = \binom{6}{4}(-2)^4 = 15 \times 16 = 240$.

Top 6 traps — SL Binomial

§3 — Generalised Binomial Series AHL 1.9

For $n \in \mathbb{Q}$ (any rational), valid for $|x| < 1$:

$$(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \dfrac{n(n-1)(n-2)}{3!} x^3 + \cdots$$

With $ax$ substitution:

$$(1 + ax)^n = 1 + n(ax) + \dfrac{n(n-1)}{2!}(ax)^2 + \cdots, \quad \text{valid for } |x| < \dfrac{1}{|a|}$$

Key differences from SL:

• Series is infinite (never terminates unless $n \in \mathbb{N}_0$).
• Only valid for $|x| < 1$ (or $|ax| < 1$).
• Generalised $\binom{n}{r} = \dfrac{n(n-1) \cdots (n-r+1)}{r!}$ — can be fractional or negative.

Generalised coefficients table

Standard series to memorise ($|x| < 1$)

§4 — The Factoring Trick: $(a + bx)^n$ when $a \neq 1$ AHL 1.9

1. Step 1: Factor out $a^n$:   $(a + bx)^n = a^n\!\left(1 + \dfrac{b}{a} x\right)^n$.
2. Step 2: Expand $(1 + u)^n$ with $u = \dfrac{b}{a} x$.
3. Step 3: State validity $|u| < 1$, i.e. $|x| < \dfrac{|a|}{|b|}$.

Example: Expand $\sqrt{4 + x}$ to 3 terms.
$\sqrt{4 + x} = 2\!\left(1 + \tfrac{x}{4}\right)^{1/2} = 2\!\left[1 + \tfrac{1}{2}\cdot \tfrac{x}{4} + \tfrac{(1/2)(-1/2)}{2}\cdot \tfrac{x^2}{16} + \cdots\right] = 2 + \tfrac{x}{4} - \tfrac{x^2}{64} - \cdots$, valid for $|x| < 4$.

Approximation by substitution

Goal: Use your series to approximate $\sqrt[k]{m}$ or $(1.03)^{-5}$, etc.

1. Expand $(1 + ax)^n$ as a series.
2. Set $1 + ax =$ target base, solve for $x$.
3. Check $|x| < \dfrac{1}{|a|}$ — must be in the valid range.
4. Substitute $x$ into the series and compute.

Top 6 traps — AHL Binomial

§5 — Partial Fractions + Binomial Series AHL 1.9

Method

1. Split into partial fractions: $\dfrac{A}{1 + ax} + \dfrac{B}{1 + bx}$.
2. Expand each using $(1 + u)^{-1} = 1 - u + u^2 - \cdots$.
3. Add the two series; collect powers of $x$.
4. Validity = the more restrictive of the two conditions.

Validity reminder

Worked example

Expand $\dfrac{5}{(1 + x)(1 - 2x)}$ to $x^2$.

Partial fractions: $\dfrac{A}{1 + x} + \dfrac{B}{1 - 2x}$.   $5 \equiv A(1 - 2x) + B(1 + x)$.
$x = -1$: $5 = 3A \Rightarrow A = \tfrac{5}{3}$.   $x = \tfrac{1}{2}$: $5 = \tfrac{3B}{2} \Rightarrow B = \tfrac{10}{3}$.

$\tfrac{5}{3}(1 - x + x^2 - \cdots) + \tfrac{10}{3}(1 + 2x + 4x^2 + \cdots)$
$= 5 + \tfrac{-5 + 20}{3} x + \tfrac{5 + 40}{3} x^2 + \cdots = 5 + 5x + 15x^2 + \cdots$.

Valid for $|x| < \tfrac{1}{2}$.

Validity table

§6 — Connection to Maclaurin & $\ln$ AHL 1.9, AHL 5.19

The generalised binomial series is the Maclaurin series of $(1 + x)^n$:

$f(x) = (1 + x)^n \Rightarrow f^{(r)}(0) = n(n-1) \cdots (n-r+1)$
$f(x) = \displaystyle\sum_{r=0}^{\infty} \dfrac{f^{(r)}(0)}{r!} x^r = \sum_{r=0}^{\infty} \binom{n}{r} x^r$.

From $(1 + x)^{-1}$, integrate to get $\ln$

$\dfrac{d}{dx} \ln(1 + x) = \dfrac{1}{1 + x} = (1 + x)^{-1} = 1 - x + x^2 - x^3 + \cdots$
$\Rightarrow \ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots$,   $|x| \leq 1$, $x \neq -1$.

§7 — GDC Skills (Paper 2) Paper 2

§8 — Exam Attack Plan SL 1.3, AHL 1.9

Worked Example — IB-Style Generalised Series

Common Student Questions