Arithmetic and geometric sequences and series, sigma notation, sum formulae, recurrence relations, increasing, decreasing and periodic sequences, and the sum to infinity of a convergent geometric series.

What this dot point is asking

OCR wants you to work with arithmetic and geometric sequences and series: find the $n$th term and the sum of $n$ terms, use sigma notation, generate sequences from recurrence relations, recognise increasing, decreasing and periodic sequences, identify when a geometric series converges, and find its sum to infinity.

The answer

Arithmetic sequences

An arithmetic sequence has a constant common difference $d$. The $n$th term and the sum are:

Geometric sequences

A geometric sequence has a constant common ratio $r$. The $n$th term and the sum are:

The sum to infinity exists only when $|r| < 1$; otherwise the terms do not shrink to zero and the series diverges. Stating this condition is often a mark in itself.

Sigma notation

$\displaystyle\sum_{r=1}^{n} u_r$ means "add the terms $u_r$ from $r = 1$ to $r = n$". For example $\sum_{r=1}^{4}(2r + 1) = 3 + 5 + 7 + 9 = 24$.

Recurrence relations and sequence behaviour

A recurrence relation defines each term from previous ones, such as $u_{n+1} = 2u_n - 3$ with $u_1 = 5$. A sequence is increasing if $u_{n+1} > u_n$ for all $n$, decreasing if $u_{n+1} < u_n$, and periodic if the terms repeat in a fixed cycle, like $u_{n+1} = \tfrac{1}{u_n}$ alternating between two values.

Examples in context

A geometric sum

Generating from a recurrence

Proving a sum using sigma results

A common Paper 1 task combines sigma notation with the standard sum formulae. The key facts are $\sum_{r=1}^{n} 1 = n$ and $\sum_{r=1}^{n} c = cn$ for a constant $c$. Combined with the arithmetic sum, these let you evaluate sums written in sigma form without listing every term.

Modelling with sequences

Sequences model real situations such as repayments, salaries that rise by a fixed percentage, or the bounce heights of a ball. A fixed yearly increase gives an arithmetic model; a fixed percentage change gives a geometric model. Identifying which model applies, then choosing the $n$th-term or sum formula, is a frequent overarching-theme OT3 application. For a geometric decay such as a ball losing a fixed fraction of its height on each bounce, the total distance travelled is found with the sum to infinity when the ratio satisfies $|r| < 1$.

Try this

Q1. Find the 20th term of the arithmetic sequence $5, 8, 11, \dots$. [2 marks]

• Cue. $u_{20} = 5 + 19 \times 3 = 62$.

Q2. A geometric series has $a = 12$, $r = \tfrac{1}{3}$. Find its sum to infinity. [2 marks]

• Cue. $S_\infty = \dfrac{12}{1 - 1/3} = 18$.