1. Sequences MATH 102 Contemporary Math S. Rook

2. Overview Section 6.6 in the textbook: – Arithmetic sequences – Geometric sequences

3. Arithmetic Sequences

4. Sequences Sequence: a list of numbers that follows some pattern. Each number in the list is referred to as a term. – Can be written as a 1, a 2, a 3, … – The n th term defines the pattern of the sequence e.g. 1, 3, 5, 7, …, 2n – 1 We will be examining two types of sequences: – Arithmetic – Geometric

5. Arithmetic Sequences Arithmetic sequence: a sequence where the difference between ANY two successive terms is equal to the same constant value – i.e. a i+1 – a i = d for every natural number i where d is the difference e.g. starts at -1 with a difference of 3 e.g. starts at 2 with a difference of ½ 5

6. Arithmetic Sequences (Continued) The formula for the n th term of an arithmetic sequence is where a 1 is the first term of the sequence and d is the difference between any two successive terms 6

7. Sums of First n Terms of an Arithmetic Sequence The n th partial sum of an arithmetic sequence is given by where a 1 is the first term and a n is the n th term Do not worry about deriving the formula – just know how to use it – e.g. What is the sum of the first 90 numbers (i.e. 1 – 90)? 7

8. Arithmetic Sequences (Example) Ex 1: For each arithmetic sequence, i) find a n and ii) find the sum from terms 1 to a n a) 2, 8, 14, 20; a 15 b) -6, -2, 2, 6; a 22

9. Arithmetic Sequences (Example) Ex 2: There is a pyramid of cans against the wall of a supermarket. There are 30 cans on the first row, 29 on the second row, 28 on the third row, and so on up to the thirtieth row where there is 1 can. How many total cans are in the stack?

10. Geometric Sequences

11. Geometric sequence: a sequence where the ratio of ANY two successive terms is equal to the same constant value for all natural numbers i where r is known as the common ratio – e.g.: a 1 = 1 and r = 2 – e.g.: a 1 = 4 and r = ½ 11

12. Geometric Sequences (Continued) The formula for the n th term of a geometric sequence is where a 1 is the first term of the sequence and r is the common ratio 12

13. Partial Sums of Finite Geometric Sequences The n th partial sum of a geometric sequence is given by where a 1 is the first term and r is the common ratio – Do not need to worry about deriving the formula – Just know how to use it e.g. Find the sum of the first 15 terms of the geometric series whose first term is 10 and second term is 5 13

14. Geometric Sequences (Example) Ex 3: For each geometric sequence, i) find a n and ii) find the sum from terms 1 to a n a) 1, 3, 9, 27; a 11 b) 3, 6, 12, 24; a 9

15. Geometric Sequences (Example) Ex 4: A ball is dropped from a height of 8 feet. The ball always bounces 7 / 8 of the distance from which it was dropped. What will be the height of the ball after the fifth bounce?

16. Arithmetic vs Geometric Sequences (Example) Ex 5: Determine whether the sequence is arithmetic, geometric, or neither: a) 1, 0, 1, 0, 1, … b)700, 750, 800, 850, … c)-8, 2, -½, …

17. Summary After studying these slides, you should know how to do the following: – Find the nth term of an arithmetic or geometric sequence – Find the sum of the first n terms of an arithmetic or geometric sequence – Differentiate between an arithmetic or geometric series Additional Practice: – See problems in Section 6.6 Next Lesson: – Linear Equations (Section 7.1)