Sequences of numbers

Notation

A sequence is an ordered list of numbers. Its terms are written $a_1, a_2, a_3, \ldots$, where the small number is the position and $a_n$ is the term in position $n$. A question may define a sequence by a formula for $a_n$, such as $a_n = 2n^2 - 1$ (so $a_3 = 17$), or by a rule that builds each term from the one before.

Arithmetic sequences

In an arithmetic sequence, each term is the previous one plus a fixed common difference $d$: 7, 11, 15, 19, … has $d = 4$. To reach $a_n$ from $a_1$ you take $n - 1$ steps, not $n$:

$$a_n = a_1 + (n - 1)d$$

The same idea links any two terms: $a_m - a_k = (m - k)d$. If you know two terms anywhere in the sequence, divide their difference by the number of steps between them to get $d$, then step to whatever term you need.

Counting terms

How many numbers are in the list 14, 17, 20, …, 98? The gap from first to last is $98 - 14 = 84$, which is $\frac{84}{3} = 28$ steps, and 28 steps join 29 numbers:

$$\text{number of terms} = \frac{\text{last} - \text{first}}{d} + 1$$

The "+1" is the fence-post rule: a fence with 28 gaps has 29 posts. For a range of whole numbers, "from 30 to 75 inclusive" contains $75 - 30 + 1 = 46$ integers. For multiples, first find the first and last multiple inside the range, then count steps.

Adding an arithmetic sequence

Pair the first term with the last, the second with the second-to-last, and so on: every pair has the same sum. So

$$\text{sum} = \text{number of terms} \times \frac{\text{first} + \text{last}}{2}$$

that is, the count times the average of the first and last terms. The sum of $1 + 2 + \cdots + n$ is $\frac{n(n+1)}{2}$; the sum of the first $n$ odd numbers is $n^2$.

Geometric sequences

In a geometric sequence, each term is the previous one times a fixed common ratio $r$: 3, 6, 12, 24, … has $r = 2$, and 80, 20, 5, 1.25, … has $r = \frac{1}{4}$. Again there are $n - 1$ steps to the $n$th term:

$$a_n = a_1 r^{n-1}$$

To tell the two kinds apart, check differences and quotients of neighbouring terms: constant differences mean arithmetic, constant quotients mean geometric. A negative ratio makes the signs alternate: with $a_1 = 3$ and $r = -2$ you get 3, $-6$, 12, $-24$, …, so terms in odd positions are positive and terms in even positions negative.

Recursive sequences

A recursive definition gives the first term (or first few) and a rule for the next term: $a_1 = 2$ and $a_n = 3a_{n-1} - 1$ for $n \ge 2$. For the first few terms no shortcut is needed; write the terms out carefully, keeping track of which position each one fills. Most errors here are index slips, such as starting the rule at $a_1$ or labelling the third term computed as $a_3$ when it is $a_4$.

Repeating sequences

Some sequences cycle: 3, $-1$, 4, 2, 3, $-1$, 4, 2, … repeats every 4 terms. To find a far-off term, divide its position by the cycle length and use the remainder, as with remainders and repeating decimals: remainder 1 means the first term of the cycle, and remainder 0 means the last. For a sum of the first $n$ terms, count the complete cycles, multiply by the sum of one cycle, and add the leftover terms.

Strategy

Write out the first few terms whenever a sequence is defined abstractly. It costs seconds and shows the pattern, the sign changes and the index convention at once. Before you answer, check one value you can verify directly, such as the term you were given.