Sequences & series

Recurrence relations, arithmetic and geometric series, the sum to infinity, and the binomial expansion.

9 min read

This topic covers sequences defined by a formula or a recurrence relation, the sums of arithmetic and geometric series, and the binomial expansion of $(1 + x)^{n}$ .

A sequence can be given by a formula for the $n$ th term, or by a recurrence relation $x_{n + 1} = f (x_{n})$ that builds each term from the previous one (you also need a starting term).

Worked example

A sequence has $x_{1} = 3$ and $x_{n + 1} = 2 x_{n} - 1$ . Find $x_{4}$ .

$x_{2} = 2 (3) - 1 = 5, x_{3} = 2 (5) - 1 = 9, x_{4} = 2 (9) - 1 = 17.$

An arithmetic series has a constant common difference $d$ . The sum of the first $n$ terms is $S_{n} = \frac{n}{2} (2 a + (n - 1) d) = \frac{n}{2} (a + l)$ , where $l$ is the last term.

A useful special case is the sum of the first $n$ natural numbers: $1 + 2 + \dots + n = \frac{n ( n + 1 )}{2} .$

Worked example

Find $1 + 2 + \dots + 100$ .

$\frac{100 \times 101}{2} = 5050.$

A geometric series has a constant ratio $r$ . The sum of $n$ terms is $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ .

If $∣ r ∣ < 1$ the terms shrink and the series converges to a finite sum to infinity: $S_{\infty} = \frac{a}{1 - r} .$

Worked example

Find the sum to infinity of $8 + 4 + 2 + 1 + \dots$

Here $a = 8$ , $r = \frac{1}{2}$ (and $∣ r ∣ < 1$ ): $S_{\infty} = \frac{8}{1 - \frac{1}{2}} = 16.$

Exam tip

A sum to infinity only exists when $∣ r ∣ < 1$ . If a question offers it, check the ratio first; if $∣ r ∣ \geq 1$ the series diverges and the answer is "no finite sum".

For positive integer $n$ , $(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots + x^{n}$ , and more generally the coefficients are $(r n) = \frac{n !}{r ! ( n - r )!}$ (read " $n$ choose $r$ ").

Worked example

Expand $(1 + x)^{4}$ .

Coefficients $(0 4), \dots, (4 4) = 1, 4, 6, 4, 1$ :

$(1 + x)^{4} = 1 + 4 x + 6 x^{2} + 4 x^{3} + x^{4} .$

Using $S_{\infty}$ when $∣ r ∣ \geq 1$ .
Confusing the $n$ th term with the sum $S_{n}$ .
Slipping on factorials: $(r n) = \frac{n !}{r ! ( n - r )!}$ , not $\frac{n !}{r !}$ .

Sequences & series

Overview

Sequences & recurrence relations

Arithmetic series

Geometric series & sum to infinity

The binomial expansion

Common mistakes