A sequence and its general term

A sequence is an ordered list of numbers that can go on without end: it has a first number, a second, a third, and so on. Each number in the list is a term of the sequence.

• The even numbers: $\displaystyle 2, 4, 6, 8, \ldots$
• The number of seats in the rows of a hall, from the first row: $\displaystyle 12, 14, 16, 18, \ldots$
• The powers of 3: $\displaystyle 1, 3, 9, 27, \ldots$

The three dots say that the sequence goes on by the same pattern.

Each term has a place in the sequence: first, second, third… The terms are written as a letter with the place below it, at its side:

• $\displaystyle a_1$ – the first term, $\displaystyle a_2$ – the second term;
• $\displaystyle a_n$ – the term in place $\displaystyle n$;
• $\displaystyle a_{n+1}$ – the term after it, and $\displaystyle a_{n-1}$ – the term before it.

The place $\displaystyle n$ is a natural number: $\displaystyle 1, 2, 3, \ldots$ – there is no term in place $\displaystyle 0$ or in place $\displaystyle 2.5$.
In the sequence $\displaystyle 2, 4, 6, 8, \ldots$: $\displaystyle {a_1}={2}$, $\displaystyle {a_3}={6}$, and $\displaystyle {a_4}={8}$.

A formula by place – the formula of the general term – gives each term from its place: substitute $\displaystyle n$, and get $\displaystyle a_n$.
In the sequence $\displaystyle {a_n}={3n-1}$:

• $\displaystyle {a_1}={3\cdot1-1}={2}$;
• $\displaystyle {a_2}={3\cdot2-1}={5}$;
• $\displaystyle {a_{10}}={3\cdot10-1}={29}$.

So a term far along is worked out without writing all the terms before it. The formula is a function defined on the natural numbers only: to each place $\displaystyle n$ it gives a number, $\displaystyle a_n$.

The first terms of $\displaystyle {a_n}={3n-1}$ can be written in a table of values:

and on a graph, where each term is a point $\displaystyle (n, a_n)$:

• The $\displaystyle x$-axis is the place $\displaystyle n$, and the $\displaystyle y$-axis the value of the term $\displaystyle a_n$.
• The graph of a sequence is separate points – a discrete graph: between $\displaystyle {n}={1}$ and $\displaystyle {n}={2}$ there is no term, so no line joins the points.

A sequence can also be described in words, and from the words we reach a formula:

• “The odd numbers, from 1”: $\displaystyle 1, 3, 5, 7, \ldots$ – each term is 1 less than twice its place: $\displaystyle {a_n}={2n-1}$.
• “The squares of the natural numbers”: $\displaystyle 1, 4, 9, 16, \ldots$ – $\displaystyle {a_n}={n^2}$.

A formula is checked on a term or two: $\displaystyle {a_4}={2\cdot4-1}={7}$, and the fourth odd number is indeed 7.

The sequence: $\displaystyle {a_n}={5n-3}$.
Calculate:

The sequence: $\displaystyle {a_n}={4n+3}$.

In which place in the sequence is the term 63?

Is 52 a term of the sequence? Explain.

Write a formula by place for each sequence:

• $\displaystyle 3, 6, 9, 12, \ldots$
• $\displaystyle 2, 4, 8, 16, \ldots$
• $\displaystyle \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$