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7. Sequences
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Integrated Math 1

7.02 Linear functions as arithmetic sequences

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Concept summary

We give some sequences special names depending on their pattern.

Arithmetic sequence

A sequence of numbers in which each consecutive pair of numbers has a common difference.

Example:

-7, -3, 1, 5, 9,\ldots

Common difference

The result of subtracting consecutive terms in an arithmetic sequence: d=a_n-a_{n-1}

Example:

-7, -3, 1, 5, 9,\ldots has d=4

The nth term, a_n of an arithmetic sequence is given by the explicit rule or general formula:

\displaystyle a_n=a_1 + d\left(n-1\right)
\bm{a_1}
The first term of the sequence
\bm{d}
The common difference
\bm{n}
The term number

For an arithmetic sequence, we can also use a recursive equation and the first term, to describe the sequence:

\displaystyle a_n=a_{n-1}+d \text{ and given } a_1
\bm{a_n}
The current term
\bm{a_{n-1}}
The previous term
\bm{d}
The common difference
\bm{a_1}
The first term of the sequence

Notations other than a_n may be used such as t_n, T_n, b_n, u_n, ...

When we represent an arithmetic sequence as a linear function whose domain is a subset of the integers, we generally use function notation and then simplify:

\displaystyle a(n)=a(1)+(n-1)d
\bm{a(n)}
The value of the nth term
\bm{a(1)}
The first term
\bm{n}
The term number
\bm{d}
The common difference, the slope of the line

Worked examples

Example 1

Consider the arithmetic sequence defined by:

a_n= 4 +3\left(n-1\right)

a

Write the recursive formula for this arithmetic sequence.

Approach

The explicit formula is of the form: a_n=a_1 + d\left(n-1\right), where a_1 is the first term and d is the common difference. We just need to match the values in the given explicit rule to find a_1 and d.

Solution

The constant when in this form is the first term, so we can say that a_1=4.

The coefficient of \left(n-1\right) is the common difference, so we can say that d=3.

Using this we can fill in the recursive formula a_n=a_{n-1}+d with a_1 to get:

a_n=a_{n-1}+3 and a_1=4.

Reflection

We could also have found the values of a_1 and d by generating terms of the sequence. For example, to find a_1:

\displaystyle a_n\displaystyle =\displaystyle 4 +3\left(n-1\right)Given
\displaystyle a_1\displaystyle =\displaystyle 4 +3\left(1-1\right)Substitute n=1
\displaystyle a_1\displaystyle =\displaystyle 4Evaluate using order of operations
b

Find the 10th term.

Approach

This means we are looking for when n=10, or in other words a_{10}. Since we have both the explicit and recursive formulas, we could use either, but the explicit will be much quicker as we don't need to find all of the terms up to a_{10} like we do with the recursive formula.

Solution

\displaystyle a_n\displaystyle =\displaystyle 4+3(n-1)Given
\displaystyle a_{10}\displaystyle =\displaystyle 4+3(10-1)Use n=10
\displaystyle =\displaystyle 31Evaluate using order of operations

The 10th term of the sequence is 31.

Example 2

Consider the arithmetic sequence:

3,\, 4.2,\, 5.4,\, 6.6,\, 7.8, \ldots

a

Write a recursive formula for the sequence.

Approach

To fill in a_n= a_{n-1}+d and a_1, we need both d and a_1.

Solution

From the given sequence, a_1= 3 and d=4.2-3=1.2.

The recursive equation for an arithmetic sequence is a_n= a_{n-1}+d

Substituting the values to the equation gives:

a_n= a_{n-1}+1.2, \, a_1=3

Reflection

Assuming that the first term in the list is a_1, we could also say: a_n= a_{n-1}+1.2, \, a_0=1.8

We could actually say any of the following and they would all represent the same sequence:

  • a_n= a_{n-1}+1.2, \, a_1=3
  • a_n= a_{n-1}+1.2, \, a_0=1.8
  • a_{n+1}= a_{n}+1.2, \, a_1=3
  • a_{n+1}= a_{n}+1.2, \, a_0=1.8

In some contexts, it we may be given a_0 instead of a_1. In particular, the y-intercept of a graphed sequence corresponds to a_0.

b

Find the next four terms in the sequence.

Approach

To find a term, our recursive rule says to "add 1.2 to the previous term", we can use this to find the next four terms.

Solution

Repeatedly adding 1.2 we get:

3,\, 4.2,\, 5.4,\, 6.6,\, 7.8,\, 9.0,\, 10.2,\, 11.4,\, 12.6\ldots

Reflection

Notice that finding the "next" terms in the sequence is efficient using the recursive rule, but finding the nth term such as a_{20} could be very tedious using this rule instead of the explicit rule.

Example 3

Consider the arithmetic sequence which has been plotted on the coordinate plane:

1
2
3
4
5
6
n
-8
-6
-4
-2
2
4
6
8
a_n
a

Identify the common difference from the graph.

Approach

We can use that the slope of the linear function which goes through the points will be the common difference.

1
2
3
4
5
6
n
-8
-6
-4
-2
2
4
6
8
a_n

Alternatively, we can create a table of values and identify it from there.

Solution

From the graph we can identify that between each term we go down 3 units, so d=-3.

Reflection

Notice that if we write the coordinates of the points from the graph in a table of values we can confirm the difference between consecutive values is d=-3.

A table of values where the first row are the following values of n: 1, 2, 3, 4, 5, and the second row is the corresponding values of a_n: 8, 5, 2, -1, -4. Below the second row arrows are used to show that the values of a_n go down by 3 each time with -3 on each arrow from one value to the next.
b

Write an explicit rule using function notation to represent the arithmetic sequence as a linear function, a(n).

Approach

The general rule will start in the form a(n)=a(1)+(n-1)d, so we need to find a(1) and d and then substitute and simplify.

Solution

1
2
3
4
5
6
n
-10
-8
-6
-4
-2
2
4
6
8
10
a_n

From the graph we can see that a(1)=8

\displaystyle a(n)\displaystyle =\displaystyle a(1)+(n-1)dGeneral rule
\displaystyle a\left(n\right)\displaystyle =\displaystyle 8+(n-1)(-3)Substitute a\left(1\right)=8 and d=-3
\displaystyle a\left(n\right)\displaystyle =\displaystyle 8-3n+3Distributive property of multiplication
\displaystyle a\left(n\right)\displaystyle =\displaystyle 11-3nCombine like terms

Reflection

1
2
3
4
5
6
n
-8
-6
-4
-2
2
4
6
8
10
12
a_n

We can also recognize that the related linear function would have slope -3 and y-intercept of (0,11), so the equation in slope-intercept form would be a(n)=-3n+11

c

Describe the domain of the linear function that is related to the sequence.

Approach

We will assume that the first term given has n=1 and can see which values of n are graphed.

Solution

The domain is all positive integers, which can be written as n \in \{1, 2, 3, 4, \ldots \} or more precisely as n \in \Z^+.

Outcomes

M1.F.BF.A.2

Define sequences as functions, including recursive definitions, whose domain is a subset of the integers. Write explicit and recursive formulas for arithmetic and geometric sequences in context and connect them to linear and exponential functions.*

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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