Math Core

A sequence is an ordered list of numbers, and many real patterns come as lists: monthly balances, stacked rows of seats, the heights of a bouncing ball. In this lesson you'll learn to describe a sequence with a formula and to write long sums compactly with sigma notation, the tools you'll use for the rest of this unit.

Sequences as functions

A sequence is a function whose inputs are the counting numbers $1, 2, 3, \dots$ Instead of writing $f(1), f(2), f(3)$ , we usually write $a_1, a_2, a_3$ . The number $a_n$ is called the $n$ th term, and $n$ is its position (or index).

Definition

Sequence

A sequence is a function whose domain is the positive integers (or a set of consecutive integers). The outputs $a_1, a_2, a_3, \dots$ are its terms. A sequence that stops is finite; one that goes on forever is infinite.

An explicit formula gives $a_n$ directly in terms of $n$ . For example, $a_n = 5n - 2$ produces

a_1 = 3, \quad a_2 = 8, \quad a_3 = 13, \quad a_4 = 18, \ \dots

The big advantage of an explicit formula is that you can jump straight to any term. The $100$ th term is $a_{100} = 5(100) - 2 = 498$ ; you don't need the $99$ terms before it.

Worked example: Generating terms

Find the first four terms and the $10$ th term of $a_n = n^2 - 3n$ .

Substitute each position for $n$ :

\begin{aligned} a_1 &= 1 - 3 = -2 \\ a_2 &= 4 - 6 = -2 \\ a_3 &= 9 - 9 = 0 \\ a_4 &= 16 - 12 = 4 \\ a_{10} &= 100 - 30 = 70 \end{aligned}

Graphing a sequence

Because a sequence is a function, you can graph it. The graph is a set of separate points $(n, a_n)$ , not a connected curve, since there is no "term number $2.5$ ". The points of $a_n = n^2 - 3n$ sit on the parabola $y = x^2 - 3x$ , which is drawn dashed below only as a guide.

The terms of aₙ = n² − 3n plotted as points (n, aₙ). The dashed parabola is a guide, not part of the sequence.Open in grapher →

Finding a formula from a pattern

Going the other way, from a list of terms to a formula, takes some detective work. Line the terms up with their positions and look for how each term depends on $n$ .

Worked example: Writing explicit formulas

Write an explicit formula for each sequence.

(a) $\dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dots$

The numerator matches the position, and the denominator is one more than the position. So $a_n = \dfrac{n}{n+1}$ .

(b) $-1, 4, -9, 16, -25, \dots$

Ignoring signs, the terms are $1, 4, 9, 16, 25$ : the perfect squares $n^2$ . The signs alternate, starting negative. The factor $(-1)^n$ is $-1$ when $n$ is odd and $1$ when $n$ is even, which is exactly this pattern. So $a_n = (-1)^n n^2$ .

Check: $a_3 = (-1)^3 \cdot 9 = -9$ . ✓

Tip

To make signs alternate, use $(-1)^n$ if the first term is negative, and $(-1)^{n+1}$ if the first term is positive.

Sigma notation

A series is the sum of the terms of a sequence. Writing out a long sum is tiring, so mathematicians use the Greek capital letter sigma, $\Sigma$ , to mean "add up."

Definition

Sigma notation

\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n

The variable $k$ is the index of summation. The number below $\Sigma$ is the lower limit (where $k$ starts) and the number above is the upper limit (where $k$ stops). You substitute each integer from the lower limit to the upper limit and add the results.

The index doesn't have to be $k$ (you'll also see $i$ , $j$ or $n$ ), and it doesn't have to start at $1$ . The number of terms in $\displaystyle\sum_{k=m}^{p}$ is $p - m + 1$ . For instance, $\displaystyle\sum_{k=3}^{10}$ has $10 - 3 + 1 = 8$ terms, not $7$ .

Worked example: Evaluating sums

(a) Evaluate $\displaystyle\sum_{k=1}^{5} (2k + 1)$ .

(2 + 1) + (4 + 1) + (6 + 1) + (8 + 1) + (10 + 1) = 3 + 5 + 7 + 9 + 11 = 35.

(b) Evaluate $\displaystyle\sum_{j=2}^{5} j^2$ .

Start at $j = 2$ , not $1$ :

2^2 + 3^2 + 4^2 + 5^2 = 4 + 9 + 16 + 25 = 54.

Common mistake

Check the lower limit before you start adding. In part (b) above, starting at $j = 1$ would add an extra $1$ and give $55$ instead of $54$ .

Writing a sum in sigma notation

To compress a sum, find a formula for the $k$ th term, then choose limits that produce exactly the terms you need.

Worked example: Compressing a sum

Write $5 + 10 + 15 + \cdots + 60$ in sigma notation.

Each term is $5$ times its position: $5 = 5(1)$ , $10 = 5(2)$ , and so on. The last term is $60 = 5(12)$ , so $k$ runs from $1$ to $12$ :

5 + 10 + 15 + \cdots + 60 = \sum_{k=1}^{12} 5k.

Other answers are possible. For example, $\displaystyle\sum_{k=0}^{11} (5k + 5)$ lists the same terms.

Properties of sums

Sums follow the same rules as ordinary addition, which gives three handy shortcuts:

Properties of sigma notation

\sum_{k=1}^{n} c = nc \qquad \sum_{k=1}^{n} c\,a_k = c\sum_{k=1}^{n} a_k \qquad \sum_{k=1}^{n} (a_k + b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k

The first rule says that adding the constant $c$ a total of $n$ times gives $nc$ . For example, $\displaystyle\sum_{k=1}^{20} 4 = 80$ . These rules let you split a complicated sum into simpler pieces whose values you already know. If you know that $1 + 2 + \cdots + 10 = 55$ , then

\sum_{k=1}^{10} (3k - 2) = 3\sum_{k=1}^{10} k - \sum_{k=1}^{10} 2 = 3(55) - 20 = 145.

Practice

Practice 1

A sequence is defined by $a_n = 3n - 7$ . Find $a_{12}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

A sequence is defined by $a_n = (-1)^n (n + 1)$ . Find $a_5$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Evaluate $\displaystyle\sum_{k=1}^{4} (k^2 + 1)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Evaluate $\displaystyle\sum_{j=3}^{7} (10 - 2j)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Which expression equals $2 + 4 + 8 + 16 + 32$ ?

Practice 6

Write an explicit formula for $a_n$ in the sequence $2, 5, 10, 17, 26, \dots$ Use $n$ for the position.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

How many terms are in the sum $\displaystyle\sum_{k=4}^{25} (k^3 - k)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Given that $\displaystyle\sum_{k=1}^{10} k = 55$ , find $\displaystyle\sum_{k=1}^{10} (2k + 3)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.