IB Math Studies Yr 1 Name_________________________________ Date_______________

IB Math Studies Year 1 6-10 Geometric Series

Learning Goal #4: What is an Geometric series? How can I find the sum of a sequence?

Warm Up: Answer the following IB question in order to prepare for today’s lesson

Antonio started to work for a business. He earns an annual salary of 8000 euros during the first year of

employment. The company increases his salary by 5% each year.

a. What would Antonio earn in his fifth year of employment?

b. Calculate the total amount that Antonio would earn during these five years?

What would happen if we needed to calculate the amount he made in total after 20 years? 30 years? 100 years? Would there be a better way to find Antonio’s total amount after n years?

Geometric Series

A Series is the _____________ of the terms in a sequence.

o An example of a sequence: 1, 3, 9, 27, 81,…

o An example of a series: 1 + 3 + 9 + 27 + 81

Notation: sn stands for the sum of the first n terms of a sequence.

Let’s Fill in our formulas!

IB Math Studies Yr 1

Geometric Formulas How do you know when to use each formula?

Sequence

𝒖𝒏 = 𝒖𝟏𝒓𝒏−𝟏

Series #1

𝑺𝒏 =𝒖𝟏(𝒓

𝒏 − 𝟏)

𝒓 − 𝟏

Series #2

𝑺𝒏 =𝒖𝟏(𝟏 − 𝒓𝒏)

𝟏 − 𝒓

Examples:

1. The first three terms of an geometric sequence are 𝑢1 = 486, 𝑢2 = 162, 𝑢3 = 54.

a. Find the value of r, the common ratio of the sequence.

b. What is the value of the 10th term?

c. What is the sum of the first 30 terms of the sequence?

IB Math Studies Yr 1

2. Each day a runner trains for a 10 mile race. On the first day she runs 20 m, and then runs 1.5 times the

previous day’s distance, on each subsequent day.

a. Find the distance she runs on the 30th day.

b. What is the total distance she will have run in training by the end of that day? Give your answer

exactly.

3. The first term of a geometric sequence is 2 and the third term is 2.205.

a. Calculate the common ratio of the sequence.

b. Calculate the eleventh term of the sequence.

c. Calculate the sum of the first 23 terms of the sequence.

IB Math Studies Yr 1

Making Connections…

4. A hydraulic hammer drives a metal post vertically into the ground by striking the tip of the post. The

distance that the post is driven into the ground, by the nth strike of the hammer, is dn.

The distances d1, d2, d3, …, dn form a geometric sequence.

The distance that the post is driven into the ground by the first strike of the hammer, d1, is 64cm.

The distance that the post is driven into the ground by the second strike of the hammer, d2, is 48cm.

a. Find the value of the common ratio for this sequence.

b. Find the distance that the post is driven into the ground by the eighth strike of the hammer.

c. Find the total depth that the post has been driven into the ground after 10 strikes of the hammer.

IB Math Studies Yr 1 Name_________________________________ Date_______________

IB Math Studies Year 1

6-10 Homework

1. Consider the following sequence: 16, 8, a, 2, 1, …1

32

a. Write down the common ratio

b. Write down the value of a

c. Find the number of terms of the sequence.

d. Find the sum of the sequence.

2. Given the following terms of a geometric sequence: U2 = 9, U4 = 36

a. Calculate the common ratio

b. What is the value of the 1st term?

c. Find the sum of the first 25 terms.

IB Math Studies Yr 1

3. A basketball is dropped vertically. It reaches a height of 2 m on the first bounce. The height of each

subsequent bounce is 10% less than the previous bounce.

a. What height does it reach on the 8th bounce?

b. What is the total vertical distance travelled by the ball between the first and sixth time the ball hits the

ground?

4. Clara wants to buy some land. She is presented with one payment option. The option requires her to pay for the

land in 20 monthly installments. The first installment is $2500. Each installment is $200 more than the one

before.

(a) Write down the values of the second and third installments.

(b) Calculate the value of the final installment.

(c) Show that the total amount that Clara would pay for the land is $88 000.