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.

$f (1) = 4, f (n) = \frac{1}{2} \cdot f (n - 1)$

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 $100$ terms of the geometric sequence given in problem 1?

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 $1$ inch, find the side-length of the next four cubes in the tower:

$1$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$

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 $n^{th}$ cube in this sequence, using the given information.

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 $4$ cubes.

3.

Find the sum of the volumes of the first $4$ cubes in the tower.

a.

Show all of your computations.

b.

Represent the sum using summation notation.

Chau thinks the cube with a side-length of $1$ inch is too small and proposes that they use four cubes, but make the cube with a side-length of $2$ inches the top cube.

4.

Find the sum of the volumes of the $4$ cubes in this tower.

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 $n$ cubes as $S_{n}$ and realized that since Chau started with the second cube in Benji’s sequence, his sum of $n$ cubes could be represented as $8 \cdot S_{n}$ .

5.

Use the diagram to explain why Kassandra’s statement, “The volume of sand in $n$ cubes of Chau’s tower is $8$ times the volume of sand in $n$ cubes of Benji’s tower,” is true.

6.

Thinking about removing the top cube of Benji’s tower and adding the bottom cube of Chau’s tower, find an expression for $8 \cdot S_{n} - S_{n}$ .

7.

Using the expression you found in problem 6, along with the fact that $8 \cdot S_{n} - S_{n} = S_{n} (8 - 1)$ , find a formula for calculating $S_{n}$ , regardless of the number of cubes in the tower.

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 $1,600 {in}^{3}$ of sand, and each additional cube in the stack above it will have a volume that is $\frac{3}{4}$ of the volume of the cube below it.

9.

If the bottom cube will have a volume of $1,600 {in}^{3}$ , find the volume of the next four cubes in the tower:

$1,600$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$ , $\underset{―}{}$

10.

Write an expression for the volume of the $n^{th}$ cube in this sequence using the given information.

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 $n$ cubes of Benji’s tower.

12.

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

$S_{n} = a + a r + a r^{2} + a r^{3} + a r^{4} + \dots + a r^{n - 1}$

13.

Design a cube tower that uses between $8,000 {in}^{3}$ and $10,000 {in}^{3}$ of sand. Give the volume of the first cube, the common ratio between the volumes of successive cubes, and the number of cubes in the tower.

Ready for More?

Benji’s parents have put $$ 100$ into a bank account each year on his birthday, starting with his first birthday. The account earns $5 %$ annually. Benji is wondering how much money he will have on his twenty-first birthday, since each $$ 100$ deposit will have been earning interest for a different length of time.

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 $$ 100$ deposited in his account on his first birthday?

b.

How much of the total amount came from the $$ 100$ deposited in his account on his fifth birthday?

c.

How much of the total amount came from the $$ 100$ deposited in his account on his tenth birthday?

d.

How much of the total amount came from the $$ 100$ deposited in his account on his twenty-first birthday, just before he withdrew all of the money in the account?

e.

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

Takeaways

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:

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 $4 + 2 + 1 + \frac{1}{2} + \frac{1}{4}$ can be represented as $\sum_{n = o}^{4} \frac{4}{2^{n}}$ which when evaluated results in.

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.

Retrieval

1.

The height of a ball hit upward from the earth’s surface at an initial velocity of $96^{\frac{ft}{s}}$ can be modeled by the function $f (t) = - 16 t^{2} + 96 t$ .

The value of $f (t) = - 16 t^{2} + 96 t$ will increase from $0$ to some maximum value and then decrease back to $0$ as it returns to the earth’s surface. Calculate the number of seconds the ball will be in the air. (Hint: Use completing the square.)

2.

Prove $△ A B C ≅ △ G E F$

3.

Prove $△ A B C ≅ △ C D A$