1. Sequences

2. Sequence There are 2 types of SequencesArithmetic: You add a common difference each time. Geometric: Geometric: You multiply a common ratio each time.

3. Arithmetic Sequences Example: {2, 5, 8, 11, 14,...} Add 3 each time {0, 4, 8, 12, 16,...} Add 4 each time {2, -1, -4, -7, -10,...} Add –3 each time

4. Arithmetic Sequences Find the 7 th term of the sequence: 2,5,8,… Determine the pattern: Add 3 (known as the common difference) Write the new sequence: 2,5,8,11,14,17,20 So the 7 th number is 20

5. Arithmetic Sequences When you want to find a large sequence, this process is long and there is great room for error. To find the 20 th, 45 th, etc. term use the following formula: a n = a 1 + (n - 1)d

6. Arithmetic Sequences a n = a 1 + (n - 1)d Where: a 1 is the first number in the sequence n is the number of the term you are looking for d is the common difference a n is the value of the term you are looking for

7. Arithmetic Sequences Find the 15 th term of the sequence: 34, 23, 12,… Using the formula a n = a 1 + (n - 1)d, a 1 = 34 d = -11 n = 15 a n = 34 + (n-1)(-11) = -11n + 45 a 15 = -11(15) + 45 a 15 = -120

8. Arithmetic Sequences Melanie is starting to train for a swim meet. She begins by swimming 5 laps per day for a week. Each week she plans to increase her number of daily laps by 2. How many laps per day will she swim during the 15 th week of training?

9. Arithmetic Sequences What do you know? a n = a 1 + (n - 1)d a 1 = 5 d= 2 n= 15 t 15 = ?

10. Arithmetic Sequences t n = t 1 + (n - 1)d t n = 5 + (n - 1)2 t n = 2n + 3 t 15 = 2(15) + 3 t 15 = 33 During the 15 th week she will swim 33 laps per day.

11. Geometric Sequences In geometric sequences, you multiply by a common ratio each time. 1, 2, 4, 8, 16,... multiply by 2 27, 9, 3, 1, 1/3,... Divide by 3 which means multiply by 1/3

12. Geometric Sequences Find the 8 th term of the sequence: 2,6,18,… Determine the pattern: Multiply by 3 (known as the common ratio) Write the new sequence: 2,6,18,54,162,486,1458,4374 So the 8 th term is 4374.

13. Geometric Sequences Again, use a formula to find large numbers. a n = a 1 (r) n-1

14. Geometric Sequences Find the 10 th term of the sequence : 4,8,16,… a n = a 1 (r) n-1 a 1 = 4 r = 2 n = 10

15. Geometric Sequences a n = a 1 (r) n-1 a 10 = 4 (2) 10-1 a 10 = 4 (2) 9 a 10 = 4 512 a 10 = 2048

16. Geometric Sequences Find the ninth term of a sequence if a 3 = 63 and r = -3 a 1 = ? n= 9 r = -3 a 9 = ? There are 2 unknowns so you must…

17. Geometric Sequences First find t 1. Use the sequences formula substituting t 3 in for t n. a 3 = 63 a 3 = a 1 (-3) 3-1 63 = a 1 (-3) 2 63= a 1 9 7 = a 1

18. Geometric Sequences Now that you know t 1, substitute again to find t n. a n = a 1 (r) n-1 a 9 = 7 (-3) 9-1 a 9 = 7 (-3) 8 a 9 = 7 6561 a 9 = 45927

19. Sequence A sequence is a set of numbers in a specific order Infinite sequence Finite sequence

20. Sequences – sets of numbers

21. Ex. 1 Find the first four terms of the sequence First term Second term Third term Fourth term

22. Ex. 2 Find the first four terms of the sequence

23. Writing Rules for Sequences We can calculate as many terms as we want as long as we know the rule or equation for a n. Example: 3, 5, 7, 9, ___, ___,……. _____. a n = 2n + 1

24. Writing Rules for Sequences Try these!!! 3, 6, 9, 12, ___, ___,……. _____. 1/1, 1/3, 1/5, 1/7, ___, ___,……. _____. a n = 3n, a n = 1/(2n-1)