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Honors: 7. Sequences
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United States of AmericaFL
Grade 912

Honors: 7.02 Linear functions as arithmetic sequences

Lesson
Practice
Lesson

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

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

Worked examples

Example 1

Consider the arithmetic sequence defined by:

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

a

Find the first term of the sequence.

Solution

The first term of the sequence is when n=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

Reflection

We could also have matched the terms in the explicit rule to see that a_1=4 based on the format of the equation.

b

Find the common difference.

Solution

As long as we know it is an arithmetic sequence, the common difference can always be calculated by subtracting the first term from the second term. Start by finding a_2.

\displaystyle a_n\displaystyle =\displaystyle 4 +3\left(n-1\right)Given
\displaystyle a_2\displaystyle =\displaystyle 4 +3\left(2-1\right)Substitute n=2
\displaystyle a_2\displaystyle =\displaystyle 7Evaluate using order of operations

Find the common difference

\displaystyle d\displaystyle =\displaystyle a_n-a_{n-1}Equation for d
\displaystyle d\displaystyle =\displaystyle a_2-a_1Use n=2
\displaystyle d\displaystyle =\displaystyle 7-4Substitute
\displaystyle d\displaystyle =\displaystyle 3Evaluate

The common difference is 3.

Reflection

The explicit rule also shows 3 as the common difference.

c

Find the 10th term.

Approach

This means we are looking for when n=10, or in other words a_{10}.

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 an explicit formula for the sequence, a_n.

Approach

To fill in a_n= a_1+d\left(n-1\right), we need both d and a_1.

Solution

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

The general equation for an arithmetic sequence is a_n= a_1+ d\left(n-1\right)

Substituting the values to the equation gives:

a_n= 3+ 1.2\left(n-1\right)

The simplified equation is

a_n= 1.2 n + 1.8

b

Find a(11).

Approach

Using the equation from part (a), let n=11 to find the 11th term of the sequence.

Solution

\displaystyle a_n \displaystyle =\displaystyle 1.2n+1.8
\displaystyle a_{11} \displaystyle =\displaystyle 1.2\left(11\right)+1.8
\displaystyle =\displaystyle 13.2+1.8
\displaystyle =\displaystyle 15

The 11th term, a_{11}=15.

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.

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 a general rule to represent the arithmetic sequence, a_n and then simplify.

Approach

The general rule will start in the form a_n=a_1+d(n-1), 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+d(n-1)General rule
\displaystyle a_n\displaystyle =\displaystyle 8-3(n-1)Substitute a_1=8 and d=-3
\displaystyle a_n\displaystyle =\displaystyle 8-3n+3Distribute the multiplication
\displaystyle a_n\displaystyle =\displaystyle 11-3nCombine like terms

Reflection

We can always check our rule is correct by substituting in a term that is not the first term.

\displaystyle a_n\displaystyle =\displaystyle 11-3nGeneral rule
\displaystyle a_5\displaystyle =\displaystyle 11-3(5)Substitute n=5
\displaystyle -4\displaystyle =\displaystyle -4Substitute a_5=-4

Outcomes

MA.912.AR.2.5

Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine constraints in terms of the context.

MA.912.AR.10.1

Given a mathematical or real-world context, write and solve problems involving arithmetic sequences.

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