Sequences and series

What you will be able to do

• read a sequence as a function on the natural numbers, finite or infinite;
• find terms from a formula for $a_n$, or from a rule that uses earlier terms;
• write a series in sigma notation, and work out a partial sum;
• recover the terms of a sequence from its partial sums.

A list in order

Rhea saves coins in a jar. She notes how many coins are in it at the end of each day. Her list begins $0, 3, 8, 15, 24, 35, \dots$. The order matters: the third number belongs to day $3$.

A list like this is a sequence. Its numbers are its terms. We write the first term as $a_1$, the second as $a_2$, and the $n$th term as $a_n$.

A sequence is a function whose inputs are the natural numbers $1, 2, 3, \dots$. The output at position $n$ is the term $a_n$.

Rhea's terms follow a rule: $a_n = n^2 - 1$. So $a_1 = 0$, $a_2 = 3$, and $a_{10} = 99$. A rule like this gives any term at once, without the ones before it.

A sequence that stops is finite, such as the marks of one class on a test. A sequence that goes on for ever is infinite, such as the odd numbers $1, 3, 5, 7, \dots$.

Terms from a rule

Not every sequence has a neat formula. Some are built step by step, from the terms before.

The Fibonacci sequence starts with $a_1 = 1$ and $a_2 = 1$. After that, each term is the sum of the two before it:

So the terms are $1, 1, 2, 3, 5, 8, 13, 21, \dots$. A rule of this kind is called a recurrence. To find $a_{10}$, you must work through the terms in order.

Series and sigma notation

Adding the terms of a sequence gives a series. For the sequence $a_1, a_2, a_3, \dots$, the series is $a_1 + a_2 + a_3 + \dots$.

Long sums are written with the Greek letter sigma, $\sum$:

The letter $k$ runs through $1, 2, \dots, n$. For example,

The sum of the first $n$ terms is the partial sum $S_n$. So $S_1 = a_1$, $S_2 = a_1 + a_2$, and so on. For Rhea's sequence the partial sums are $0, 3, 11, 26, 50, 85, \dots$.

Terms from the partial sums

Each partial sum is the one before it, plus one new term: $S_n = S_{n-1} + a_n$. Turn this round:

So if you know every partial sum, you know every term.

A worked example

The sum of the first $n$ terms of a sequence is $S_n = 2n^2 + 3n$. Find $a_1$ and $a_n$.

1. The first term. $a_1 = S_1 = 2 + 3 = 5$.
2. The general term. $S_{n-1} = 2(n - 1)^2 + 3(n - 1) = 2n^2 - n - 1$.
3. Subtract. $a_n = (2n^2 + 3n) - (2n^2 - n - 1) = 4n + 1$.
4. Check. At $n = 1$, $4n + 1 = 5$, which agrees with $a_1$.

The terms are $5, 9, 13, 17, \dots$. Each is $4$ more than the one before. The next lesson names sequences like this.

Practise with new numbers

Try it in Python

The program makes the terms of $a_n = n^2 - 1$, builds the running totals, and then gets the terms back from the totals.

Common mistakes

• Mixing up $a_n$ and $S_n$. $a_n$ is one term. $S_n$ adds the first $n$ terms.
• Using $a_n = S_n - S_{n-1}$ at $n = 1$. There is no $S_0$ in the formula. Take $a_1 = S_1$.
• Skipping terms in a recurrence. A rule that uses earlier terms must be followed in order.
• Reading the sigma limits wrongly. $\sum_{k=1}^{5}$ adds five terms, from $k = 1$ to $k = 5$.

Remember these

• A sequence is a function on $1, 2, 3, \dots$; its $n$th term is $a_n$.
• Terms come from a formula for $a_n$, or from a recurrence.
• $S_n = a_1 + a_2 + \dots + a_n = \sum_{k=1}^{n} a_k$.
• $a_1 = S_1$, and $a_n = S_n - S_{n-1}$ for $n \geq 2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Write the first four terms of $a_n = \dfrac{n}{n + 1}$.

2. A sequence has $a_1 = 4$ and $a_n = 3a_{n-1} - 2$ for $n > 1$. Find $a_4$.

3. Work out $\sum_{k=1}^{4} k^2$.

4. The partial sums of a sequence are $S_n = n^2 + n$. Find $a_7$.

5. Is the list of a city's daily rainfall in June a finite or an infinite sequence?