# Lesson 9 Summing Up Sand Castles Solidify Understanding

## Jump Start

### 1.

Find the sum of the first 5 terms of this geometric sequence. Show all of your work.

### 2.

Write the explicit equation for the geometric sequence given in problem 1.

### 3.

About how large do you think the sum would be for the first

## Learning Focus

Find the sum of a geometric sequence.

How can I add up the terms in a geometric sequence efficiently, without adding them one by one?

How can I design a tower of stacked cubes whose volume grows geometrically to fit within a given constraint on the total volume?

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

Benji, Chau, and Kassandra like to test out their sand castle designs by using clay models. Each part of the castle can be formed independently and then assembled into a model they can take to the sand sculpting competition.

One feature they have planned for the sand castle is to make a tower out of stacked cubes. Each cube will have side-lengths that are twice the length of the sides of the cube above it.

### 1.

If the top cube will have a side-length of

Benji, Chau, and Kassandra haven’t decided how many cubes to include in the stack that will form the tower.

### 2.

Write an expression for the side-length of the

Benji recalls that they need to calculate the volume of the sand that will be used to create the tower of stacked cubes. He suggests they start by calculating the volume of a tower with

### 3.

Find the sum of the volumes of the first

#### a.

Show all of your computations.

#### b.

Represent the sum using summation notation.

Chau thinks the cube with a side-length of

### 4.

Find the sum of the volumes of the

#### a.

Show all of your computations.

#### b.

Represent the sum using summation notation.

Because they haven’t decided how many cubes to include in the tower, Kassandra has been trying to find a formula for calculating the total volume of sand without having to add the volumes of each cube separately. Benji’s and Chau’s computations have helped her to think about that. Kassandra represented the sum of

### 5.

Use the diagram to explain why Kassandra’s statement, “The volume of sand in

### 6.

Thinking about removing the top cube of Benji’s tower and adding the bottom cube of Chau’s tower, find an expression for

### 7.

Using the expression you found in problem 6, along with the fact that

### 8.

Verify that your formula works for the sum of the volume of the four cubes in Benji’s tower.

Pause and Reflect

Benji thinks the cubes grow too quickly to make a pleasant looking tower and proposes that they start with a base cube that contains

### 9.

If the bottom cube will have a volume of

### 10.

Write an expression for the volume of the

Chau thinks the volume of the bottom cube is too large and wants to create a tower that starts with the base cube being the second cube in Benji’s tower.

### 11.

Using Kassandra’s strategy of comparing Benji’s and Chau’s towers from problems 5–8, find a formula for the sum of the volume of the first

### 12.

In general, find a formula for the following sum of terms:

### 13.

Design a cube tower that uses between

## Ready for More?

Benji’s parents have put

Benji plans to withdraw all of the money in the bank account on his twenty-first birthday, immediately after his parents make the twenty-first deposit, to make a down payment on a car.

#### a.

If Benji withdraws all of the money in the account on his twenty-first birthday, how much of the total amount came from the

#### b.

How much of the total amount came from the

#### c.

How much of the total amount came from the

#### d.

How much of the total amount came from the

#### e.

How much money will be in the account on Benji’s twenty-first birthday?

## Takeaways

The sum of

where:

## Adding Notation, Vocabulary, and Conventions

Summation notation can be used to represent the sum of a sequence of terms.

For example, this sum for

When the terms being added form a geometric sequence, then the sum of the terms is called a .

## Vocabulary

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

## Lesson Summary

In this lesson, we learned how to find the sum of the terms in a geometric sequence without needing to add each term in the sequence one at a time.

### 1.

The height of a ball hit upward from the earth’s surface at an initial velocity of

The value of

### 2.

Prove

### 3.

Prove