# Lesson 6 Compounding the Problem Develop Understanding

## Learning Focus

Understand how different factors affect the amount earned from compound interest.

Determine the features of a new function:

Which factor makes the most difference in the amount of money earned in a savings account: the interest rate, the number of compounding periods per year, or the number of years invested?

## Open Up the Math: Launch, Explore, Discuss

Part I: As an enterprising young mathematician, you know that your superior knowledge of mathematics will help you make better decisions about all kinds of things in your life. One important area is money $$$. So, you’ve been contemplating the world and wondering how you could maximize the money that you make in your savings account.

You’re young and you haven’t saved much money yet. As a matter of fact, you only have $100, but you really want to make the best of it. You like the idea of compound interest, meaning that the bank pays you interest on all the money in your savings account, including whatever interest that they had previously paid you. This sounds like a very good deal. You even remember that the formula for compound interest is exponential. Let’s see, it is:

Where

### 1.

If your saving account pays a generous

### 2.

How much money would be in the account at the end of

It seems like the more compounding periods in the year, the more money that you should make. The question is, does it make a big difference?

### 3.

Compare the amount of money that you would have after

It turns out that the value you found in your compounding problem is 100 times a very famous irrational number, named

### 4.

It is fairly typical for savings accounts to be compounded monthly. Compare the amount of money in two savings accounts after

### 5.

Use technology to compare the graphs of the two accounts. What conclusions would you draw about the effect of changing the number of compounding periods on a savings account?

Part II

Since

### 6.

Make a prediction about the graph of

### 7.

Create a table and a graph and describe the mathematical features of

## Ready for More?

If you could invest your

#### a.

Changing the number of compounding periods from quarterly to daily?

#### b.

Changing the interest rate from

#### c.

Changing the amount invested to

## Takeaways

Features of

## Adding Notation, Vocabulary, and Conventions

## Vocabulary

- continuous compound interest
- explicit equation
- irrational number
- recursive equation
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson, we explored the effects of changing the number of compounding periods in an investment that earns compound interest. In the course of this exploration, we found the number

### 1.

Use first and second differences to identify the type of function represented by the pattern in each table. If the pattern in the table is linear, write both the explicit and the recursive equations. If the pattern is quadratic, write only the recursive equation.

#### a.

**Table 1:**

#### b.

**Table 2:**

### 2.

Fill in the blanks using what you know about logarithms.

I know that