Lesson 7 Step It Up, Sum It Up Practice Understanding

Jump Start

Martina’s grandparents put $$ 50$ in an envelope on the day Martina was born, and they add $$ 50$ to the envelope on her birthday each year. They plan to give the envelope to Martina on her $21 st$ birthday. Martina is hoping she can convince her grandparents to give her the money sometime sooner. Martina and her friends have each created a graphical representation showing visually how much money will be in the account at any point in time before her $21 st$ birthday.

1.

Write the equation of the function represented by each of the friends’ graphs.

a.

Equation for Martina's Graph:

b.

Equation for Jorge’s Graph:

c.

Equation for Nguyen’s Graph:

2.

What is useful about each person’s graph in terms of representing the context?

a.

Martina’s graph:

b.

Jorge’s graph:

c.

Nguyen’s graph:

3.

In your opinion, which graph represents the context best? Why?

Martina’s friends think she should talk her grandparents into depositing the money into a bank account instead of keeping it in an envelope. Not only would the money be safer kept in a bank account, it would also earn interest. To illustrate the growth of the money in the bank for Martina’s grandparents, they assumed the account would earn $4 %$ interest annually. They decided to focus only on the initial deposit of $$ 50$ and show its value during the $21$ years. Here are their representations:

4.

Write the equation of the function represented by each of the friends’ graphs.

a.

Equation for Martina’s Graph:

b.

Equation for Jorge’s Graph:

c.

Equation for Nguyen’s Graph:

5.

What is useful about each person’s graph in terms of representing the context?

a.

Martina’s graph:

b.

Jorge’s graph:

c.

Nguyen’s graph:

6.

In your opinion, which graph represents the context best? Why?

Learning Focus

Represent contexts using piecewise functions and step functions.

Write an expression for the value of a financial account where regular payments or deposits are made at equal intervals of time and find the value of the account.

How can I model contexts such as mortgage payments and annuities as the sum of several terms forming a geometric sequence?

How do I quickly sum up the terms in a geometric series?

Open Up the Math: Launch, Explore, Discuss

Martina likes the idea of having her grandparents deposit their annual birthday gift in a bank account earning $4 %$ annually, but realizes she and her friends have only represented one deposit into the account—the deposit made on the day she was born. She would like to represent all of the deposits into the account. The type of account that consists of equal deposits over equal increments of time is referred to as an annuity.

Here is what Martina has written to represent the initial deposit, assuming the money will remain in the bank until her $21 st$ birthday: $f (t) = 50 {(1.04)}^{21}$ . She has calculated the value of this deposit to be worth $$ 113.94$ on her $21 st$ birthday.

1.

The $$ 50$ deposit made on Martina’s first birthday will only grow for $20$ years. Find the value of this deposit on her $21 st$ birthday.

2.

Find the value of the deposit made on Martina’s second birthday when she is $21$ years old.

3.

Find the value of the deposit made on Martina’s third birthday when she is $21$ years old.

Using this approach, Martina will need to calculate the value of each deposit on her $21 st$ birthday individually, and then sum up each of the terms to get a total amount. Since the individual terms form a geometric sequence, the sum of the terms creates a geometric series.

Martina is also interested in being able to find the value of the account if she is able to take out the money at any time before her $21 st$ birthday. (She knows the bank will pay her interest even if she takes the money out of the account on a day other than her birthday.) That is, she wants to treat the value of the account as a function of time, rather than just a sum of specific values. Martina would like a more efficient strategy for calculating the value of the account at any instant in time during the $21$ years, without having to recalculate each of the terms individually.

4.

Martina’s first attempt was to write a piecewise function for the account. Show the first 4 sub-function rules, and the last sub-function rule, that would be part of a piecewise function description of the money in her account. Use the variable $t$ to represent the amount of time the money has been in the account and n to represent the year in which the money is removed.

$f (t) = {\begin{cases} 50 {(1.04)}^{t}, 0 \leq t < 1 \\ \underset{―}{} \\ \underset{―}{} \\ \underset{―}{} \\ ⋮ \\ \underset{―}{} \end{cases}$

Martina recognizes that each sub-function rule is an expression with more and more terms. Since she doesn’t know how many terms to include, Martina’s next step was to represent the sum of all of these terms using summation notation. Recall that summation notation is of the form $\sum_{t = 1}^{n} f (t)$ .

5.

Use summation notation to represent the amount of money in the bank after $n$ years.

Writing a geometric series using summation notation reminds Martina that she developed a formula for the sum of a geometric series in her Geometry class. She pulls out her Geometry notes to remind her of where the formula came from and how it works.

Here are her takeaway notes:

The sum of $n$ terms of a geometric sequence, $S_{n} = a + a r + a r^{2} + a r^{3} + a r^{4} + \dots + a r^{n - 1}$ is given by the formula:

$S_{n} = \frac{a \cdot r^{n} - a}{r - 1}$

where $a$ is the first term in the sequence, $a \cdot r^{n - 1}$ is the last term, and $r$ is the common ratio between the terms.

She also found this diagram and her notes about how the formula was derived for the specific case given in the diagram:

The volume of the tower on the right is $8$ times the volume of the tower on the left; and
The tower on the right contains a cube whose volume is given by $1 \cdot 8^{n}$ , but omits the cube whose volume is $1$ .
From these two observations I can write the equation: $8 \cdot S_{n} - S_{n} = 8^{n} - 1$
I can then solve this equation for $S_{n}$ : $S_{n} = \frac{8^{n} - 1}{8 - 1}$

6.

Martina’s notes relate to a geometric context and a specific case of a geometric sequence where the constant term is $1$ and $r = 8$ . Using Martina’s notes as a guide, explain why the sum of the terms in any geometric series represented by a polynomial of the form $S_{n} = a + a r + a r^{2} + a r^{3} + \dots + a r^{n - 1}$ is given by the formula $S_{n} = \frac{a \cdot r^{n} - a}{r - 1}$ .

7.

Using this formula, find the value of Martina’s account if she waits until the final deposit is made on her $21 st$ birthday.

8.

Find the value of Martina’s account if her grandparents allow her to remove the money after the deposit is made on her $18 th$ birthday.

9.

Find the value of Martina’s account if she waits until the deposit is made on her $21 st$ birthday, but the annual percentage rate for the account is $5 %$ instead of $4 %$ .

Ready for More?

Piecewise functions for which the sub-functions are defined over equal increments can often be written with a single rule using the greatest integer function $f (x) = [x]$ . When $x$ is an integer, the value of the greatest integer function is the integer. When $x$ is not an integer, the value of the function is the greatest integer less than $x$ . For example: $[2.3] = 2$ , $[4.75] = 4$ , $[- 2.873] = - 3$ and $[π] = 3$ .

1.

In the Jump Start, Nguyen’s graph could be represented as a piecewise function using a set of sub-functions. Rewrite this function as a single statement using the greatest integer function.

2.

Suppose Martina’s grandparents had decided to put $$ 100$ in an envelope on the day she was born and then add $$ 100$ to the account on her even birthdays, but no money on her odd birthdays, with the last deposit being made on her $20 th$ birthday.

a.

Sketch a graph of the amount of money in the account until Martina’s $21 st$ birthday using this scenario.

b.

Write an equation for this graph using the greatest integer function.

Takeaways

An annuity is an account in which . The value of an annuity can be represented by .

The value of an annuity can be found by:

Vocabulary

annuity
geometric series
summation notation
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to find the value of an annuity—an account into which equal deposits are made at equal intervals of time. We modeled annuities as geometric series and calculated the sum of the terms of the geometric series using a formula derived in a previous lesson.

Retrieval

Twenty-eight students took an exam containing $30$ questions. The data showing the scores of the $28$ students are represented in a dot plot, a boxplot, and a histogram. The teacher used the data to calculate the following statistics.

Mean: about $23.6$

Median: $24.5$

Mode: $25$

Standard deviation: about $5.2$

1.

Which representation makes the mode most visible?

2.

Which representation includes the median as one of its features?

3.

Describe how you can tell in each representation that most of the students scored higher than the mean.

a.

dot plot:

b.

boxplot:

c.

histogram:

4.

Is the score of $5$ on the exam an outlier? Explain.

5.

The graph shows the product of a linear function and a trigonometric function. Describe the features of the graph that helps you to see the two factors that make up the product.

6.

What features of this graph help you to see that the function is the absolute value of a sine function?