Sequences and Series

1. Sequences and Series Algebra 2

2. Vocabulary • Sequence • Series • Term • Domain • Range • Infinite • Finite • Summation (Sigma) Notation

3. Sequences and Series • Find the sum 1 + 2 + 3 + . . . + 9 + 10 • Find the sum 1 + 2 + 3 + . . . + 98 + 99 + 100 • Find the sum 1 + 2 + 3 + . . . + 999 + 1000 • Find the sum 2 + 4 + 6 + . . . + 398 + 400

4. Sequences and Series • What is a sequence? • What is a series? • What is the difference? • What is arithmetic, what is geometric? • What is sigma (summation) notation? • What formulas are important?

5. What is an Arithmetic Sequence? • An arithmetic sequence is when there is a common difference between consecutive terms. • The common difference is constant. It is written as d. • Example: 2, 5, 8, 11, 14,… • a1 = 2, a2 = 5, a3 = 8, a4 =11, a5 = 14 • The common difference would be 3. • The way you find “d” is by: 5 – 2 = 3, 8 – 5 = 3, 11 – 8 = 3, 14 – 11 = 3

6. Rule for an Arithmetic Sequence • The nth term of an arithmetic sequence with first term a1and common difference “d” is:

7. Writing a Rule for the nth Term • Example: 2, 5, 8, 11, 14, … • The common difference is 3 • Use the equation : an= a1+ (n – 1)d • a1= 2 • an= 2 + (n – 1)3 • an= 2 + 3n – 3 • an= 3n – 1  That is your rule for the nth term

8. Johann Carl Friedrich Gauss • Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a Germanmathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

9. Johann Carl Friedrich Gauss • Another famous story has it that in primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels. • Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050.

10. Find the Sum of a Finite Arithmetic Series • The expression formed by adding the terms of an arithmetic sequence is called an arithmetic series. The equation for the sum of a finite arithmetic series is:

11. Find the Sum of a Finite Arithmetic Series (cont.) • Example: 2 + 5 + 8 + 11 + 14 … • Find the sum of the first 20 terms. • First off, you have to find a formula for the nth term. We did this on our first example which came out to be : an= 3n – 1 • Now you plug in 20for n: a20 = 3(20)-1=60-1=59 • Now use to find the sum of first 20 terms  That is your answer

12. Application #1 • A well-drilling company charges $15 for drilling the first foot, $15.25 for the second foot, $15.50 for the third foot, and so on. • This means that it would cost $45.75 for the company to drill 3 feet. • How much would it cost for the company to drill a 100-foot well?

13. Application #2 • The first row of a concert hall has 25 seats, and each row after that has one more seat than the row before it. There are 32 rows of seats. • Suppose each seat in rows 1 through 11 of the concert hall costs $24, each seat in rows 12 through 22 costs $18 and each seat in rows 23 through 32 costs $12. • How much money does the concert hall take in for a sold out event?

14. Geometric Sequences • With arithmetic sequences we added something each time • With geometric sequences we multiply by something each time

15. Geometric Sequences Here are some geometric sequences: We multiplied by 2 each time. We multiplied by –½ each time. How do you find the ratios? Sometimes you can just look. The ratio is 4. But usually you don’t get that lucky . . . OMG, how do you find the ratio here?

16. Geometric Sequences Geometric Sequences have a common ratio. The common ratio is named r. To find r we simply divide any term by its previous term. Can you find the ratio from the sequence in the previous slide? Find the ratio for the following sequence:

17. Geometric Sequences A geometric sequence is kind of like exponential growth or decay. You will multiply by a common ratio to find the next term. To find the general formula for a geometric sequence: Find the general rule for the following:

18. Geometric Series To find the sum of a finite geometric series, use the following formula: Evaluate the following:

19. Geometric Infinite Series • Suppose everyday money was deposited into an account for you. • On the first day you received $100, on the next day you received $50, on the third day you received $25, and this pattern continued for the for the rest of your life and let’s assume you will live longer than Yoda (which I think is like more than 800 years) • How much money would be in your account? • What is happening to the amount of money being deposited? • About how much money is being deposited on the 30th day? • How do you think this will affect the balance?

20. Geometric Infinite Series • The sum of an infinite series can be found with the following formula: Evaluate the following: