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7. Sequences
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Integrated Math 1

7.04 Applications of sequences

Lesson
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Lesson

Concept summary

The explicit rules for arithmetic and geometric sequences can be applied to many real-world scenarios.

Recall that for an arithmetic sequence:

\displaystyle a_n=a_1 + d\left(n-1\right)
\bm{a_n}
The nth term of the sequence
\bm{a_1}
The first term of the sequence
\bm{d}
The common difference
1
2
3
4
n
-5
-4
-3
-2
-1
1
2
3
4
5
a_n

An arithmetic sequence will have a common difference - related to linear growth and adding.

Examples of arithmetic sequences include changing by equal amounts and simple interests earned from an investment or charged in a loan.

Recall for a geometric sequence:

\displaystyle a_n=a_1 \cdot r^{n-1}
\bm{a_n}
The nth term of the sequence
\bm{a_1}
The first term of the sequence
\bm{r}
The common ratio
1
2
3
4
n
2
4
6
8
10
12
14
16
18
20
22
24
a_n

A geometric sequence will have a common ratio - related to exponential growth and multiplying.

Examples of geometric sequences include the growth or decay of a certain population and compound interests earned from an investment or charged in a loan.

In some real-world contexts involving geometric sequences, we may want to use an initial term of a_0 instead of a_1. In these cases, we can alter the explicit formula slightly and use the following formula:

\displaystyle a_n=a_0 \cdot r^{n}
\bm{a_n}
The (n+1)th term of the sequence
\bm{a_0}
The first term of the sequence
\bm{r}
The common ratio

Worked examples

Example 1

Ethan is playing a new game on his phone. After successfully playing his first game on day 1, he was awarded with 25 diamonds. The game then rewards him 3 diamonds for each consecutive day he plays after day 1.

a

Determine if the sequence for the number of diamonds he has after n days of consecutive play is arithmetic or geometric.

Approach

To identify whether the given problem involves an arithmetic or geometric sequence, we should determine first if there is a common difference or a common ratio.

Solution

The number of diamonds increases by 3 each day, so there is a common difference of d=3. Therefore, the problem involves an arithmetic sequence.

b

Determine the number of diamonds Ethan will receive when after playing 6 consecutive days.

Approach

We can use the formula for the nth term of an arithmetic sequence, a_n=a_1 + d\left(n-1\right) to write an explicit rule that models this scenario.

Solution

First we must find an explicit rule for the number of diamonds Ethan gets each day on his game:

\displaystyle a_n\displaystyle =\displaystyle a_1 + d\left(n-1\right)General rule for an arithmetic sequence
\displaystyle a_n\displaystyle =\displaystyle 25 + 3\left(n-1\right)Substitute a_1 = 25 and d=3
\displaystyle a_n\displaystyle =\displaystyle 25+3n-3Distribute
\displaystyle a_n\displaystyle =\displaystyle 3n+22Combine like terms

Now we can use our explicit rule to find a_6.

\displaystyle a_{n}\displaystyle =\displaystyle 3n+22Explicit rule
\displaystyle a_{6}\displaystyle =\displaystyle 3(6)+22Substitute n=6
\displaystyle a_{6}\displaystyle =\displaystyle 40Evaluate

Ethan will earn 40 diamonds on the 6th day.

c

Find the number of consecutive days Ethan will need to play to earn 160 diamonds.

Approach

We can use the rule from part (b): a_{n}=3n+22, to find the number of days, n, it will take to get a_n=160 diamonds.

Solution

\displaystyle a_{n}\displaystyle =\displaystyle 3n+22Explicit rule
\displaystyle 160\displaystyle =\displaystyle 3n+22Substitute a_n=160
\displaystyle 138\displaystyle =\displaystyle 3nSubtract 22
\displaystyle 46\displaystyle =\displaystyle nDivide by 3
\displaystyle n\displaystyle =\displaystyle 46

It will take Ethan 46 days to earn 160 diamonds.

Reflection

Alternatively, we can find the number of additional days after day 1 in a more conceptual way without using the equation:

\displaystyle 160-25\displaystyle =\displaystyle 135Number of diamonds Ethan needs to earn after day 1
\displaystyle 135 \div 3\displaystyle =\displaystyle 45Number of days Ethan needs to play after day 1
\displaystyle 45+1\displaystyle =\displaystyle 46Total number of days Ethan needs to play

Example 2

A ball is dropped onto the ground from a height of 8 \text{ m}. On each bounce, the ball reaches a maximum height of 60\% of its previous maximum height.

a

Determine if the sequence of the heights of each bounce is arithmetic or geometric.

Approach

To identify whether the given problem involves an arithmetic or geometric sequence, we should determine first if there is a common difference or a common ratio.

Solution

In the problem, the height of the previous bounce is multiplied 60\% or 0.6 to get the next height. So the common ratio is r=0.6. So the sequence is geometric.

b

Determine the height that the ball reaches after the 4th bounce.

Approach

In part (a), we determined that the problem involves a geometric sequence with a common ratio of r=0.6. We also know that the ball had an initial height of 8\text{ m}, so a_0=8. We are using a_0 since the initial height occured before the first bounce.

Now, we can use the formula for the nth term of a geometric sequence, a_n=a_0 \cdot r^{n}, to write an explicit rule that models this scenario.

Solution

\displaystyle a_n\displaystyle =\displaystyle a_0 \cdot r^{n}General rule for a geometric sequence
\displaystyle a_n\displaystyle =\displaystyle 8 \cdot 0.6^{n}Substitute the first term and common ratio

Now we can use our explicit rule to find the height of the 4th bounce, where n=4:

\displaystyle a_{n}\displaystyle =\displaystyle 8 \cdot 0.6^{n}Explicit rule
\displaystyle a_{4}\displaystyle =\displaystyle 8 \cdot 0.6^{4}Substitute n=4
\displaystyle a_{4}\displaystyle =\displaystyle 1.0368Evaluate

The height of the 4th bounce will be 1.0368\text{ m}.

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.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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