2412 Pre-Calculus Chapter 12 Section 1 Mathematical Patterns

1. 2412 Pre-CalculusChapter 12 Section 1Mathematical Patterns

2. Infinite Sequence A function whose domain is the set of positive integers a1, a2, a3, a4, . . . Each of these is a termof the sequence.

3. Example 1: Write the first 4 terms of the sequence an = 4n – 2 a1 = 4(1) – 2 = 2 a2 = 4(2) – 2 = 6 a3 = 4(3) – 2 = 10 a4 = 4(4) – 2 = 14

4. Example 2: Write the first 4 terms of the sequence an = 8 – 2n a1 = 8 – 21 = 6 a2 = 8 – 22 = 4 a3 = 8 – 23 = 0 a4 = 8 – 24 = -8

5. Example 3: Write the first 4 terms of the sequence: a1 = 6/-2= -3 a2 = 12/1= 12 a3 = 18/6 = 3 a4 = 24/13

6. Equations for a sequence Think about what is happening to the top and to the bottom.

7. Factorial – a “Special” Sequence This sequence starts with the given number and multiplies it by all positive integers less than the number (stop at 1). Notation: n!

8. Example: Find:

9. Example:

10. Example:

11. Recursive Sequence A sequence that starts with a given value and builds from there.

12. A college has 24,000 students. Each May 27% graduate and in August there are 6850 new Freshmen. What will the population be in 5 years? 370 increase 144 increase

14. Finite Sums Add the integers from 1 to 10 1 + 2 + 3 + . . . + 8 + 9 + 10 11

15. Sigma Notation Stop when you reach this integer Put the integers in this equation Sum the answers Put in this integer first 2(1) + 1 = 3 2(4) + 1 = 9 2(2) + 1 = 5 2(5) + 1 = 11 2(3) + 1 = 7 2(6) + 1 = 13 2(7) + 1 = 15

16. How much money would you have if youput $2 in on day 1, $4 on day 2, $6 on day 3, and continued for 100 days? 10,100 2 + 4 + 6 + . . . + 196 + 198 + 200 =

17. Write in Sigma Notation