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Algebra 2: Lesson 12-6 Geometric Series Word Problems
Learning Goals:
1. How do we use the geometric series formula when working with word problems?
DO NOW: Answer the following question in order to prepare for today’s lesson.
1. What are the formulas for exponential growth and exponential decay? What do the variables represent in
each formula?
2. A computer valued at $6500 depreciates at the rate of 14.3% per year.
a. Write a function that models the value of the computer after n years.
b. Find the value of the computer after three years.
3. A geometric sequence has a first term of 36 and a common ratio of .
a. Write a geometric series formula,
, to represent this sum over n terms.
b. Use this formula to find the sum of the first 15 terms.
GEOMETRIC SEQUENCES WITH PERCENTAGES
**GEOMETRIC SEQUENCE FORMULA**
To find the any term of a geometric sequence:
where
= the nth term,
a1 = the first term
r = the common ratio


If a question refers to a percent, this means you are dealing with a geometric sequence.
When given a percent, the common ratio is the percent remaining of the previous term.
Example: Identify the common ratio in each situation.
a. A certain water filtration system
can remove 70% of the
contaminants each time a
sample of water is passed
through it.
b. John’s salary earns an increase
of 4% each successive year.
c. A basketball is dropped
vertically. The height of each
subsequent bounce is 90% of
the previous bounce.
1. A fan is running at 10 revolutions per second. After it is turned off, its speed decreases at a rate of 75% per
second.
Find an explicit formula for the sequence that represents the number of revolutions after n seconds.
2. Suppose you drop a tennis ball from a height of 15 feet. After the ball hits the floor, it rebounds to 85% of its
previous height.
Write an explicit formula for the sequence.
GEOMETRIC SERIES WORD PROBLEMS
Geometric Series
Sn = the sum of n terms,
a1 = the first term,
r = the common ratio
3. George has taken a job with a starting salary of $50,000 and receives an annual raise of 2%.
Write a geometric series formula,
, for George's total earnings over n years.
Use this formula to find George's total earnings for his first 12 years of working, to the nearest cent.
4. The first swing of a pendulum travels 40 centimeters. Each subsequent swing travels 95% as far as the
previous swing.
Write a geometric series formula,
, for the pendulum’s total distance over n swings.
Use this formula to find the pendulum's total distance after the 30th swing, to the nearest tenth of a centimeter.
PUTTING IT ALL TOGETHER
5. A car with an original price of
depreciates by
each year.
a. Write an explicit formula for the price of the car after n years.
b. Write a recursive formula the price of the car after n years.
c. Write a geometric series formula,
, for the car’s total price over n years.
6. Kristin wants to increase her running endurance. According to experts, a gradual mileage increase of 10% per
week can reduce the risk of injury. If Kristin runs 8 miles in week one, which expression can help her find the
total number of miles she will have run over the course of her 6-week training program?
1)
2)
3)
4)