Sequences and series

A sequence is an ordered list of numbers. A series is the result of adding terms of a sequence. That distinction is the starting point for the whole topic:

$$2,5,8,11,\ldots \qquad \text{is a sequence,}$$

whereas

$$2+5+8+11+\cdots \qquad \text{is a series.}$$

At A-level, you learn to describe terms, recognise arithmetic and geometric structure, calculate finite and infinite sums, use sigma notation, expand binomials and build models from repeated change.

Prerequisites

You should be able to:

• substitute into formulae and simplify expressions
• solve linear and quadratic equations
• work with powers, fractions and percentages
• find an nth term from a simple pattern
• use a scientific calculator accurately

Review foundation sequences, linear equations or the laws of indices if these skills are uncertain.

Four ideas that organise the topic

1. Position matters

Write $u_n$ for the term in position $n$. The subscript is a label, not multiplication: $u_5$ means the fifth term, not $u\times5$.

A sequence can be given explicitly, with $u_n$ written directly in terms of $n$:

$u_n=3n-1.$

Then $u_1=2$, $u_2=5$ and $u_{20}=59$.

Alternatively, a sequence can be given recursively, with each term defined from an earlier term:

$u_1=2,\qquad u_{n+1}=u_n+3.$

The initial value is essential. The rule $u_{n+1}=u_n+3$ describes infinitely many sequences until a starting term is supplied.

Worked example 1: generate a recursive sequence

Given

$u_1=10,\qquad u_{n+1}=\frac12u_n+3,$

find $u_2$, $u_3$ and $u_4$.

Apply the rule one step at a time:

$$\begin{aligned} u_2&=\frac12(10)+3=8,\\ u_3&=\frac12(8)+3=7,\\ u_4&=\frac12(7)+3=6.5. \end{aligned}$$

Do not substitute $n=2$ into the right side. It contains $u_n$, not $n$. Study sequences and recurrence relations for iteration, increasing and decreasing sequences, and long-term behaviour.

2. Arithmetic means add; geometric means multiply

An arithmetic sequence has a constant difference $d$:

$u_{n+1}-u_n=d.$

If its first term is $a$, then

$\boxed{u_n=a+(n-1)d}.$

A geometric sequence has a constant ratio $r$:

$\frac{u_{n+1}}{u_n}=r,$

where the ratio is defined. If its first term is $a$, then

$\boxed{u_n=ar^{n-1}}.$

The exponent or multiple is $n-1$ because no step has occurred at the first term.

Sequence	Test	Type
$4,9,14,19,\ldots$	differences are $5$	arithmetic
$80,40,20,10,\ldots$	ratios are $\frac12$	geometric
$1,4,9,16,\ldots$	neither is constant	neither

Worked example 2: find an unknown term

An arithmetic sequence has $u_7=23$ and $u_{18}=67$. Find its first term and common difference.

Using $u_n=a+(n-1)d$:

$a+6d=23,\qquad a+17d=67.$

Subtracting gives

$11d=44,$

so $d=4$. Then $a+24=23$, giving $a=-1$.

The sequence is therefore

$-1,3,7,11,\ldots$

and $u_n=-1+4(n-1)=4n-5$. Continue with arithmetic sequences and series.

Worked example 3: distinguish a term from a sum

For the geometric sequence with $a=3$ and $r=2$, find the sixth term and the sum of the first six terms.

The term is

$u_6=3(2^5)=96.$

The finite geometric sum is

$$\begin{aligned} S_6&=\frac{a(r^6-1)}{r-1}\\ &=\frac{3(2^6-1)}{2-1}\\ &=189. \end{aligned}$$

Thus $u_6=96$ but $S_6=189$. Confusing $u_n$ with $S_n$ is one of the most common errors in this topic. Learn finite sums, infinite sums and convergence in geometric sequences and series.

3. Infinite does not automatically mean infinite in value

For a geometric series,

$a+ar+ar^2+\cdots,$

the terms approach zero when $|r|<1$. Its partial sums then approach the finite limit

$\boxed{S_\infty=\frac{a}{1-r}},\qquad |r|<1.$

For example,

$12+6+3+\frac32+\cdots$

has $a=12$ and $r=\frac12$, so

$S_\infty=\frac{12}{1-1/2}=24.$

The condition $|r|<1$ is not optional. If $r=2$, the terms grow. If $r=-1$, they oscillate. In neither case do the partial sums approach one fixed number.

Worked example 4: form an infinite series from a recurring decimal

Express $0.272727\ldots$ as a fraction.

Separate the repeated blocks:

$0.272727\ldots=0.27+0.0027+0.000027+\cdots.$

This is geometric with

$a=\frac{27}{100},\qquad r=\frac1{100}.$

Therefore

$$S_\infty =\frac{27/100}{1-1/100} =\frac{27}{99} =\frac3{11}.$$

This works because $|r|<1$.

4. Notation compresses structure

The Greek capital letter sigma, $\Sigma$, means “sum”. For example,

$\sum_{k=1}^{5}(2k+1)$

means substitute $k=1,2,3,4,5$ and add:

$3+5+7+9+11=35.$

The lower value gives the starting index and the upper value gives the final index. The number of terms is

$5-1+1=5.$

The index letter is temporary, so

$\sum_{k=1}^{5}(2k+1)=\sum_{j=1}^{5}(2j+1).$

Study sigma notation before using standard sum formulae or manipulating indexed expressions.

Binomial expansions are finite series

For a non-negative integer $n$, the binomial theorem states

$$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r.$$

The coefficient

$\binom nr=\frac{n!}{r!(n-r)!}$

counts the ways to choose which $r$ of the $n$ brackets contribute a factor of $b$.

Worked example 5: find one term without expanding everything

Find the coefficient of $x^3$ in $(2-x)^6$.

The general term is

$T_{r+1}=\binom6r2^{6-r}(-x)^r.$

An $x^3$ term requires $r=3$. Its coefficient is

$\binom632^3(-1)^3=20\cdot8\cdot(-1)=-160.$

Notice that $r=3$ gives the fourth term because the first term corresponds to $r=0$. Learn this method in the binomial expansion for positive integer powers, then extend it to rational powers, where convergence restrictions become important.

Recommended learning route

Follow this order if you are learning the topic for the first time.

1. Sequences and recurrence relations develops explicit and recursive definitions, iteration and sequence behaviour.
2. Arithmetic sequences and series covers nth terms, finite sums and equations involving unknown terms.
3. Geometric sequences and series covers constant ratios, finite sums, convergent infinite series and applications.
4. Sigma notation expresses and manipulates sums compactly.
5. The binomial expansion for positive integer powers uses combinations, Pascal’s triangle and the general term.
6. The binomial expansion for rational powers introduces infinite expansions, approximations and validity intervals.
7. Sequences and series in modelling applies repeated additive or multiplicative change and tests the assumptions of a model.

Diagnostic self-check

1. Terms and notation

Given $u_n=n^2-3n$, find $u_5$ and $u_{n+1}$.

Answer

$u_5=5^2-3(5)=10.$

Replace every $n$ by $n+1$:

$$\begin{aligned} u_{n+1}&=(n+1)^2-3(n+1)\\ &=n^2-n-2. \end{aligned}$$

The expression $u_{n+1}$ does not mean $u_n+1$.

2. Arithmetic or geometric?

Classify $162,54,18,6,\ldots$ and find its eighth term.

Answer

Each term is multiplied by $\frac13$, so the sequence is geometric with $a=162$ and $r=\frac13$.

$u_8=162\left(\frac13\right)^7=\frac2{27}.$

3. A finite sum

Find $5+9+13+\cdots+81$.

Answer

First count the terms:

$5+(n-1)4=81,$

so $n=20$. Then

$S_{20}=\frac{20}{2}(5+81)=860.$

The last term alone does not tell you the number of terms.

4. Convergence

Find the sum to infinity of $7-2.1+0.63-\cdots$.

Answer

The ratio is $r=-0.3$. Since $|r|=0.3<1$, the series converges:

$S_\infty=\frac7{1-(-0.3)}=\frac{70}{13}.$

Use $1-r$, including the sign of $r$.

5. Sigma notation

Evaluate

$\sum_{r=2}^{6}(3r-1).$

Answer

There are $6-2+1=5$ terms:

$5+8+11+14+17=55.$

Common misconceptions

• A sequence is not a series. $u_n$ denotes one term; $S_n$ usually denotes a sum.
• The nth term is not found by extending a pattern repeatedly. Use a formula when $n$ is large.
• Arithmetic and geometric describe different operations. Arithmetic sequences add a constant; geometric sequences multiply by a constant.
• An infinite sum needs a convergence check. Use $S_\infty=\frac{a}{1-r}$ only when $|r|<1$.
• Indices must be counted inclusively. From $r=p$ to $r=q$ there are $q-p+1$ terms.
• A model has a domain. A geometric population model may predict non-integer values or unrealistic indefinite growth. Interpret the mathematics in context.

What mastery looks like

You are secure when you can move between a list, an nth-term formula, a recurrence relation, sigma notation and a contextual model without treating them as unrelated techniques. At the highest level, you should also justify convergence, choose an efficient formula rather than merely recall one, and check whether an algebraically valid answer is meaningful in its model.