1. Objectives Understand the difference between a finite and infinite series. Write and evaluate an arithmetic series. 1. 2.

2. Understand the difference between Finite and Infinite Series The sum of the terms of a sequence.Series: Finite Sequences & Series:Can be counted and totaled (has an end). Infinite Sequences & Series:Cannot be counted and totaled (does not have an end). Finite Sequence 6, 9, 12, 15, 18 Infinite Sequence 3, 7, 11, 15,... Finite Series 6 + 9 + 12 + 15 + 18 Infinite Series 3 + 7 + 11 + 15...

3. Use the finite sequence 2, 11, 20, 29, 38, 47. Write the related series, and then evaluate. Related Series:147 Example 1: Writing & Evaluating a Series 2 + 11 + 20 + 29 + 38 + 47 = Example 1: Practice 1.21, 18, 15, 12, 9, 6, 3 2. 100, 99, 98,..., 95 3. 17.3, 19.6, 21.9, 24.2, 26.5 21 + 18 + 15 + 12 + 9 + 6 + 3; 84 100 + 99 + 98 + 97 + 96 + 95; 585 17.3 + 19.6 + 21.9 + 24.2 + 26.5; 109.5

4. Sum of a Finite Arithmetic Series The sum S n of a finite arithmetic series a 1 + a 2 + a 3 +... a n is where a 1 is the 1 st term, a n is the nth term, & n is the # of terms. Arithmetic Series A series whose terms form an arithmetic sequence. When a sequence has many terms, or when you know only the 1 st and last terms of the sequence, you can use a formula to evaluate the related series quickly.

5. Use the formula to evaluate the series related to the following sequence: 5, 7, 9, 11, 13 45 Example 2: Evaluating an Arithmetic Series = = 5252 (5 + 13) 2.5(18) =

6. Each sequence has 8 terms. Evaluate each related series. Example 2: Practice 5. 5, 13, 21,..., 61 6. 1,765, 1,414, 1,063,..., -692 4. 4,292 32 264

7. Objectives Discover the sum of the terms of an arithmetic series. Interpret summation notation and be able to rewrite as a series. 1. 2.

8. Using Summation Notation You can use the summation symbol  to write a series. Then you can use limits to indicate how many terms you are adding. Limits are the least and greatest integral value of n.  3 n = 1 (5n + 1) upper limit, greatest value of n. explicit formula for the sequence lower limit, least value of n.

9. The explicit formula for the sequence is ____. Use summation notation to write the series 3 + 6 + 9 +... for 33 items. Example 3: Writing a Series in Summation Notation 1 3 = 3 2 3 = 6 3 3 = 9 3n 3 + 6 + 9 +... + 99 =  33 3n 1 The lower limit is ___. The upper limit is ___. 33 1

10. Use summation notation to write each arithmetic series for the specified number of terms. Example 3: Practice 13. 2 + 4 + 6 +... ; n = 4 15. 5 + 6 + 7 +... ; n = 7 17. 7 + 14 + 21 +... ; n = 15  2n4 n = 1  (n + 4) 7 n = 1  7n15

11. To expand a series from summation notation, substitute each value of n into the explicit formula and add the results. Example 4: Finding the Sum of a Series Find the first and last terms of the series. a. Use the series  3 (5n + 1). n = 1 Find the number of terms in the series. 33 Since the values of n are 1, 2, & 3, there are ___ terms. 3 b. Term #1 =(5n + 1) =5(1) + 1 =6 Term #3 =(5n + 1) =5(3) + 1 =16 Evaluate the series.c. = = = 5(1) + 1 +5(2) + 1 +5(3) + 1 6 +11 +16  3 (5n + 1) n = 1

12. Example 4: Practice For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. 5, 1, 9; 25 19. 21. 23.  5 (2n - 1) n = 1  8 (7 - n) n = 3  10 n = 2 4n 3 6, 4, -1; 9

13. Example 5: Real-World Application If Mary goes to the prom with Brian, he will be able to practice what he learned about sequences and series in Alg. 2 class. 1 + 4 + 16 + 64 + 256. 1 date 4 hours of fun 16 cups of fruit punch The price of the tickets How many times Bobby tried to ask her but chickened out. Questions 1.Is this series arithmetic, geometric, or neither? geometric 2.Kayla, will you go to the prom with Bobby? ?