1. 12.3 Geometric Series

2. Series
A series is the expression formed by adding the terms of a sequence.
If the sequence is geometric and has a finite number of terms, it is called a finite geometric series.
The sum of a geometric series is: 𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓
𝑎 is the first term
𝑟 is the ratio (expressed as a fraction or decimal) (never a percentage)
𝑛 is the number of terms

3. Proof of the Formula to find the Sum of a Geometric Series
To find a general formula for the sum of a finite geometric series with n terms, begin by writing the series as 𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +…+𝑎 𝑟 𝑛−1 Multiplying by r gives: 𝑟𝑆 𝑛 =𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +…+𝑎 𝑟 𝑛 Subtracting 𝑆 𝑛 −𝑟𝑆(𝑛) gives: 𝑎+𝑎𝑟+𝑎 𝑟 2 +…+𝑎 𝑟 𝑛−1 − 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +…+𝑎 𝑟 𝑛 = 𝑎−𝑎 𝑟 𝑛 So 𝑆 𝑛 −𝑟𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 . Factoring out the GCF, 𝑆(𝑛), on the left side and the GCF, 𝑎, on the right side gives: 𝑆 𝑛 (1−𝑟) =𝑎(1− 𝑟 𝑛 ) Dividing by 1−𝑟 on both sides gives: 𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

4. Ex:
Find a, r, and n for the geometric series 𝑎=3 𝑟=2 𝑛=5

5. Note:
If some of the terms of the terms of a finite geometric series have been replaced by an ellipsis, as in …+1458, you obviously can’t count the number of terms.
One way to deal with this situation is to generate the missing terms by using the common ratio, which in this case is 3.
The next term after 18 is 3(18) = 54, and repeatedly multiplying by 3 to generate successive terms gives , so the series has 7 terms.

6. Note (cont.):
Another way to find the number of terms in …+1458 is to recognize that the nth term in a geometric sequence is 𝑎 𝑛 =𝑎 (𝑟) 𝑛−1 (last class).
For the series …+1458 whose nth term is 2 (3) 𝑛−1 , find n as follows:
1458=2 (3) 𝑛−1
729= 3 𝑛−1
3 6 = 3 𝑛−1
6=𝑛−1
𝑛=7

7. Ex:
Find the sum of the finite geometric series …

8. Ex:
Find the sum of the finite geometric series. 1 2 − …− 1 256

9. On Your Own:
Find the sum of the finite geometric series. 5−15+45…−1215
To series should go 5−15+45− −1215 since the ratio is -3.
Since 𝑎=5, 𝑟=−3, and 𝑛=6, the sum is
𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 =𝟓 𝟏− (−𝟑) 𝟔 𝟏−(−𝟑) =5 1− =5 − =5 −182 =−910

10. Writing Exponential Growth/Decay Models
𝑦=𝑎 (1−𝑟) 𝑡 exponential decay
𝑎 is the initial amount
𝑟 is the rate (always expressed as a decimal)
1+𝑟 is the growth factor
1−𝑟 is the decay factor
𝑡 is the time

11. Ex:
You deposit $500 in an account that pays 8% annual interest compounded yearly.
Write an equation to represent the account balance after t years.
Exponential Growth
𝑦=𝑎 (1+𝑟) 𝑡
𝑦=500 (1+0.08) 𝑡
𝑦=500 (1.08) 𝑡
What is the account balance after 6 years?
𝑦=500 (1.08) 6 𝑦=
𝑦=$793.44

12. The population triples every year so the growth factor is 3.
Ex:
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
What is the percent of increase each year?
Exponential Growth
The population triples every year so the growth factor is 3.
1+𝑟=3
𝑟=2
Percent of increase = 200%
What is the equation to represent the population of rabbits?
𝑦=20 (3) 𝑡
What is the population after 5 years?
𝑦=20 (3) 5
𝑦=20 243
𝑦=4860

13. Ex (cont.):
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
Graph the growth of the rabbit population. t y

14. Ex: A new car costs $23,000. The value decreases by 15% each year.
Write an exponential model to represent the car’s value after t years.
Exponential Decay
𝑦=𝑎 (1−𝑟) 𝑡
𝑦=23,000 (1−0.15) 𝑡
𝑦=23,000 (0.85) 𝑡
To the nearest dollar, how much will the car be worth after 4 years?
𝑦=23,000 (0.85) 4
𝑦=23,000( )
𝑦=$12,006

15. Ex (cont.):
A new car costs $23,000. The value decreases by 15% each year.
Graph the decay of the car’s value. t y

16. Ex:
Jenna has been renting the same apartment for 6 years. Her monthly rent in her first year was $600. Each year after the first, her rent increased by 5.5% of the rent from the previous year. What is the total amount that Jenna paid in rent over the first 6 years of living in her apartment?
We want to find the sum of a geometric series. We will use the formula 𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 with 𝑎= =7200, 𝑟=1.055, and 𝑛=6. So
𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 =𝟕𝟐𝟎𝟎 𝟏− (𝟏.𝟎𝟓𝟓) 𝟔 𝟏−𝟏.𝟎𝟓𝟓 =𝟕𝟐𝟎𝟎 𝟏−𝟏.𝟑𝟕𝟖𝟖𝟒𝟐𝟖𝟎𝟔𝟕𝟔𝟏𝟖𝟗𝟎𝟔𝟐𝟓 −𝟎.𝟎𝟓𝟓 =𝟕𝟐𝟎𝟎 −𝟎.𝟑𝟕𝟖𝟖𝟒𝟐𝟖𝟎𝟔𝟕𝟔𝟏𝟖𝟗𝟎𝟔𝟐𝟓 −𝟎.𝟎𝟓𝟓 =𝟕𝟐𝟎𝟎 𝟔.𝟖𝟖𝟖𝟎𝟓𝟏𝟎𝟑𝟐𝟎𝟑𝟒𝟑𝟕𝟓 =$𝟒𝟗,𝟓𝟗𝟑.𝟗𝟕

17. Ex:
In 1995, the average annual salary for women in the U.S. was $30,000, and that average has steadily increased by 8% per year. If Charlotte started working in 2000, and assuming there are no other raises, how much money did she earn over her first 10 years in the workforce?
Since we don’t know how much Charlotte made in 2000, we have to use the Explicit formula for geometric sequences to find out. So
𝑎 𝑛 = 𝑎 1 (𝑟) 𝑛−1 =30,000 (1+0.08) 5−1 =30,000 (1.08) 4 =30, =$40,814.67
Now we want to find the sum of a geometric series. We will use the formula 𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 with 𝑎=40,814.67, 𝑟=1.08, and 𝑛=10. So
𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 =𝟒𝟎,𝟖𝟏𝟒.𝟔𝟕 𝟏− (𝟏.𝟎𝟖) 𝟏𝟎 𝟏−𝟏.𝟎𝟖 =𝟒𝟎,𝟖𝟏𝟒.𝟔𝟕 𝟏−𝟐.𝟏𝟓𝟖𝟗𝟐𝟒𝟗𝟗𝟕𝟐𝟕𝟐𝟕𝟖𝟔𝟔𝟗𝟖𝟐𝟒 −𝟎.𝟎𝟖 =𝟒𝟎,𝟖𝟏𝟒.𝟔𝟕 −𝟏.𝟏𝟓𝟖𝟗𝟐𝟒𝟗𝟗𝟕𝟐𝟕𝟐𝟕𝟖𝟔𝟔𝟗𝟖𝟐𝟒 −𝟎.𝟎𝟖 =𝟒𝟎,𝟖𝟏𝟒.𝟔𝟕 𝟏𝟒.𝟒𝟖𝟔𝟓𝟔𝟐𝟒𝟔𝟓𝟗𝟎𝟗𝟖𝟑𝟑𝟕𝟐𝟖 =$𝟓𝟗𝟏,𝟐𝟔𝟒.𝟐𝟕

18. On Your Own:
For each amount and annual interest rate below, use the formula for the sum of a finite geometric series to determine whether depositing that amount of money into a bank account at the beginning of every year for four years will result in an account balance of at least $5000 at the time the fourth deposit is made. Assume that interest is compounded annually. $1000; 9% interest Yes No $1200; 5% interest Yes No $1150; 7% interest Yes No $1100; 7.5% interest Yes No $1050; 8.5% interest Yes No
Answer:
No (you would have only $4,573.13)
Yes (you would have $5,172.15)
Yes (you would have $5,105.93)
No (you would have only $4,920.21)
No (you would have only $4,766.49)

19. Conversions
Although an interest rate is typically expressed as an annual rate, it can be converted to a rate for other periods of time.
Ex: An annual interest rate of 𝑟% results in a monthly interest rate of 𝑟 12 %.
In general, if interest is earned n times per year, an annual interest rate of 𝑟% is divided by n.

20. Ex:
The balance B, in dollars, after t years of an investment that earns interest compounded annually is given by the function 𝐵 𝑡 =1500 (1.045) 𝑡 . To the nearest hundredth of a percent, what is the monthly interest rate for the investment?
0.38%
3.68%
4.50%
69.59%
Answer: (A)
Since the formula for exponential growth is 𝑦=𝑎 (1+𝑟) 𝑡 , we know that 𝑎=1500 and, to find r, we solve 1+𝑟= That means the growth rate is .045 (or 4.5%) for each year. To find the monthly interest rate, we divide this percent by 12 (since there are 12 months in a year). So 4.5% 12 =0.375%.

21. Ex:
After t days, the mass m, in grams, of 100 grams of certain radioactive element is given by the function 𝑚 𝑡 = (0.97) 𝑡 . To the nearest percent, what is the weekly decay rate of the element? 3%
19%
21%
81%
Answer: (C)
Since the formula for exponential decay is 𝑦=𝑎 (1−𝑟) 𝑡 , we know that 𝑎=1 and, to find r, we solve 1−𝑟=0.97. That means the decay rate is .03 (or 3%) for each day. To find the weekly decay rate, we multiply this percent by 7 (since there are 7 days in a week). So 3×7=21%.

22. Ex:
The population of a colony of bacteria doubles every 8 hours. If the number of bacteria starts at 80, the population P after t hours is given by 𝑃 𝑡 =80∙ 2 𝑡 8 . What is the equivalent form of 𝑃 𝑡 that shows the approximate hourly growth factor for the bacteria population?
𝑃 𝑡 =80∙ 𝑡
𝑃 𝑡 =80∙ 𝑡
𝑃 𝑡 =80∙ 1.25 𝑡
𝑃 𝑡 =80∙ 𝑡
Answer: (A)
We want to find out how the equation would look like if we only wanted t in exponent. So we want to get rid of the /8 part. Using the properties of exponents Power to a Power rule, we can rewrite the equation as 𝑃 𝑡 =80∙ 𝑡 . Now we just have to find Use your calculator and type 2 𝑦 𝑥 (1/8) = and you will get

23. Ex:
An investment of $5,200 earns 6% interest compounded annually. The value V of the investment after t years is given by the function 𝑉 𝑡 = 5200 (1.06) 𝑡 . Use properties of exponents to rewrite the given function as another function that shows the approximate monthly interest rate of the investment. What is the approximate monthly rate? Explain.
Since we know t represents years, and we want it to be monthly, we will write t/12 instead. Now we have to simplify 𝑉 𝑡 =5200 (1.06) 𝑡 Using the properties of exponents Power to a Power rule, we can rewrite the equation as 𝑉 𝑡 = 𝑡 . Now we just have to find Use your calculator and type 𝑦 𝑥 (1/12) = and you will get So the new equation, where t is in months, is 𝑉 𝑡 =5200 (1.05) 𝑡 . The approximate monthly rate is 5% since we solve 1+𝑟=1.05.

24. Ex:
Niobe is saving for a down payment on a new car, which she intends to buy a car a year from now. At the end of each month, she deposits $200 from her paycheck into a dedicated savings account, which earns 3% annual interest that is applied to the account balance each month. After making 12 deposits, how much money will Niobe have in her savings account?
We want to find the sum of a geometric series. We will use the formula 𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 with 𝑎=200, 𝑟= =1.0025, and 𝑛=12. So
𝑺 𝒏 =𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓 =𝟐𝟎𝟎 𝟏− (𝟏.𝟎𝟎𝟐𝟓) 𝟏𝟐 𝟏−𝟏.𝟎𝟎𝟐𝟓 =𝟐𝟎𝟎 𝟏−𝟏.𝟎𝟑𝟎𝟒𝟏𝟓𝟗𝟓𝟕 −𝟎.𝟎𝟎𝟐𝟓 =𝟐𝟎𝟎 −𝟎.𝟎𝟑𝟎𝟒𝟏𝟓𝟗𝟓𝟕 −𝟎.𝟎𝟎𝟐𝟓 =𝟐𝟎𝟎 𝟏𝟐.𝟏𝟔𝟔𝟑𝟖𝟐𝟕𝟕 =$𝟐,𝟒𝟑𝟑.𝟐𝟖