The binomial expansion of $(a + b)^n$ for positive integer $n$ using binomial coefficients, and sequences and series described by sigma notation and recurrence.

What this dot point is asking

CCEA wants you to expand $(a + b)^n$ for a positive integer $n$ using binomial coefficients, find a particular term or coefficient without writing the whole expansion, and describe sequences and series using sigma notation and recurrence relations. The binomial expansion is a recurring AS skill and the foundation for the general binomial series at A2.

The answer

Binomial coefficients and Pascal's triangle

The binomial expansion for positive integer n

To find a single term, use the general term $\binom{n}{r} a^{n-r} b^{r}$ and choose the $r$ that gives the required power. This avoids expanding the whole bracket when you only need one coefficient.

Sequences and series

A sequence is an ordered list of terms; a series is their sum. A sequence can be defined in two ways. A formula for the $n$th term gives any term directly, for example $u_n = 2n + 1$ produces $3, 5, 7, \dots$, so you can jump straight to $u_{50}$ without listing the earlier terms. A recurrence relation instead defines each term from the previous one, such as $u_{n+1} = u_n + 3$ with a stated first term $u_1 = 4$, which you must generate step by step. Sigma notation writes a sum compactly: $\sum_{r=1}^{n} u_r$ means add the terms from $r = 1$ to $r = n$, so $\sum_{r=1}^{4} r^2 = 1 + 4 + 9 + 16 = 30$. The lower and upper numbers are the first and last values of the counter.

Worked example: a term independent of x

Examples in context

Example 1. Approximating a power

Putting $x$ small in $(1 + x)^n$ gives a quick approximation: $(1.02)^4 \approx 1 + 4(0.02) + 6(0.02)^2 = 1.08240$, since higher terms are tiny. This is why the binomial expansion underpins numerical estimation.

Example 2. A recurrence model

A savings account with $u_{n+1} = 1.03 u_n + 200$ adds $3\%$ interest and a $\pounds200$ deposit each year. The recurrence generates the balance term by term, showing how sequences model real growth before the formal series work at A2. Starting from $u_1 = 1000$, the next balance is $u_2 = 1.03(1000) + 200 = 1230$, then $u_3 = 1.03(1230) + 200 = 1466.90$, and so on, so a recurrence is the natural description when each year depends on the one before.

Example 3. Choosing a coefficient quickly

In an exam you are often asked for just one term, such as the coefficient of $x^4$ in $(3 - x)^9$. Rather than expand all ten terms, use the general term $\binom{9}{r}(3)^{9-r}(-x)^r$ and set $r = 4$: $\binom{9}{4}(3)^5(-1)^4 = 126 \times 243 = 30618$. The general-term method turns a long expansion into a single calculation, which is why CCEA rewards it.

Try this

Q1. Write down the coefficients in the expansion of $(a + b)^4$. [1 mark]

• Cue. $1, 4, 6, 4, 1$.

Q2. Find the coefficient of $x^2$ in $(3 + x)^4$. [3 marks]

• Cue. $\binom{4}{2}3^2 = 6 \times 9 = 54$.

Q3. A sequence is defined by $u_{n+1} = 2u_n - 1$, $u_1 = 3$. Find $u_3$. [2 marks]

• Cue. $u_2 = 2(3) - 1 = 5$, then $u_3 = 2(5) - 1 = 9$.