1. S EQUENCES MCC9 ‐ 12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

2. W HAT IS A SEQUENCE ? A sequence is a function whose domain is a set of consecutive whole numbers. If a domain is not specified, it is understood that the domain starts with 1. The domain represents the position of each term in the sequence. The range values are the output or the values at each term. A sequence can be specified by an equation, or a rule.

3. I NFINITE VS F INITESEQUENCES Infinite Sequences go on forever and use the ellipses 2, 4, 6, 8, 10,…, 8, 16, 24, 32, …, A Finite Sequence has a set number of terms 2, 4,6, 8, 10, 12 8, 16, 24, 32, 40

4. P ARTS OF A S EQUENCE Term Numbers Term 1 4 325 12 11 975 1 st Term or A 1 +2 Common Difference (d) The amount or # of units added to or subtracted from a term to get to the next term.

5. C ONTINUED You can use all of these parts of a sequence to write 2 different formulas that can be used to find terms in that sequence. (These are the equations of that sequence.)

6. R ECURSIVE F ORMULA The first type is a Recursive Formula. Uses the term immediately in front of the term you are looking for (labeled ) and the common difference (labeled d). The recursive formula looks like this: A n-1 Term you want Term in front of the one you want Common difference

7. W RITING THE R ECURSIVE F ORMULA When asked to write a recursive formula for a sequence, you only need to replace the d in the formula. You leave everything else exactly the same. EXAMPLE: 1. -3, -2, -1, 0, 1, 2, 3… The recursive formula for this sequence is: 2. 3, 6, 9, 12, 15… The recursive formula for this sequence is:

8. E XPLICIT F ORMULA The 2 nd type of formula is know as the explicit formula. This can be used to find a term without needing to know the term in front of the one you are looking for. The explicit formula looks like this: Term you want Common difference Term # of The term you are looking for First term of the sequence

9. E XAMPLE OF E XPLICIT F ORMULA 1. 7, 5, 3, 1, -1, -3…. To write the explicit formula, you need to first identify the common difference (d) and the first term (A1). d = -2A1 = 7 Now, just replace those letters in the formula with the numbers you have.

10. C ONTINUED 2. -6, -2, 2, 6, 10… d = 4 A1 = -6 3. 1, 5, 9, 13, 17… d = 4 A1 = 1

11. Y OU T RY Write the explicit and recursive formulas for the following sequences. 1. 5, 12, 19, 26, 33….. 2. -3, -1, 1, 3, 5….. 3. 10, 30, 50, 70, 90….

12. Y OU T RY Write the explicit and recursive formulas for the following sequences. 1. 5, 12, 19, 26, 33….. 2. -3, -1, 1, 3, 5….. 3. 10, 30, 50, 70, 90…. 1. R) A n = A n-1 + 7E) A n = 7(n-1)+5 2. R) A n = A n-1 + 2E) A n = 2(n-1)-3 3. R) A n = A n-1 + 20E) A n = 20(n-1)+10

13. F INDING T ERMS IN A S EQUENCE When you are asked to find terms of a sequence, you need the explicit formula. Example: Find the 50 th term of the sequence. 5, 12, 19, 26, 33…. First, find the d and the A1. d = 7 A1 = 5 Next, write the formula: Finally, replace “n” with the term you want (50 in this case) and solve.

14. E XAMPLE 2 2. -3, -1, 1, 3, 5… Find the 50 th term. d = 2 A1 = -3 Formula: Solve:

15. E XAMPLE 3 3. 10, 30, 50, 70, 90… Find the 50 th Term d=20 A1=10 Formula: Solve:

16. Y OU T RY Write the explicit formula for the sequences below, then use the formula to find the indicated term. You must show your work! 1. 9, 18, 27, 36, 45… Find 10 th term 2. -10, -3, 4, 11, 18… Find 15 th term 3. 4, 1, -2, -5, -8… Find 25 th term

17. Y OU T RY Write the explicit formula for the sequences below, then use the formula to find the indicated term. You must show your work! 1. 9, 18, 27, 36, 45… Find 10 th term 2. -10, -3, 4, 11, 18… Find 15 th term 3. 4, 1, -2, -5, -8… Find 25 th term 1. Formula A n = 9(n-1)+9 A 10 = 9(10-1)+9 9(9)+9 =81+9= 90 2. Formula A n = 7(n - 1) - 10 A 15 = 7(15 - 1)-10 =7 (14) – 10 =98 – 10 = 88 3. Formula A n = -3(n-1)+4 A 25 = -3(25 – 1)+4= -68

18. A RITHMETIC VS G EOMETRIC Arithmetic - Sequences are created with a common difference in the terms 2, 4, 6, 8, 10, 12 common difference is 2 Geometric - Sequences are created with a common ratio 2, 4, 8, 16, 32 common ratio is 2