1. Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12,... (2). 1, 5, 9, 13,... Write an explicit formula for: (3). 10, 7, 4, 1,... (5). -6, -4, -2,...

2. Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12,... a n = a n-1 + 2 with a 1 = 6 (2). 1, 5, 9, 13,... a n = a n-1 + 4 with a 1 = 1 Write an explicit formula for: (3). 10, 7, 4, 1,... a n = 10 + (n – 1)(-3) (5). -6, -4, -2,... a n = -6 + (n – 1)(2)

3. Arithmetic Series Section 2.11B Standard: MM2A3 d Essential Question: Can I evaluate and describe an arithmetic series?

4. Vocabulary: Series: the expression that results when the terms of a sequence are added together Sigma notation: another name for summation notation, which uses the Greek letter, sigma, written ∑ Arithmetic series: the expression formed by adding the terms of an arithmetic sequence, denotes by S n

5. Investigation 1: A series is the expression that results when the terms of a sequence are added together. Using your calculator, find the value of each series: (1). 3 + 7 + 11 + 15 + 19 = _________ (2). 14 + 34 + 54 + 74 = _________ 55 176

6. To indicate a particular sum, the notation S n can be used. S indicates summation and n identifies which terms are to be added. Thus S 2 tells us to add the first two terms of the sequence. Calculate each indicated sum. (3). 5 + 9 + 13 + 17 + 21 a. S 2 = ______ b. S 4 = ______ c. S = ______ ( 4). 12 + 9 + 6 + 3 + 0 + (– 3) + (– 6) + (– 9) a. S 2 = ______ b. S 4 = ______ c. S = ______ 144465 2130 12

7. A series is sometimes written using sigma notation. Sigma is a Greek letter and is used to indicate a sum. The sigma notation is read as the sum of all terms a i for i from 1 to n. The sigma notation can also be written using the explicit formula for a sequence.

8. Example: tells us to add the first 3 terms of a sequence where a i = 2i + 5. Calculate the value of the three terms: a 1 = 2 + 5 = 7 a 2 = 4 + 5 = 9 a 3 = 6 + 5 = 11 Now add the terms together: = 7 + 9 + 11 = 27.

9. Find each sum: (5). = ____ + ____ + ____ + ____ = _____ (6). = ___ + ___ + ___ + ___ + ___ = ____ 12963 30 0123 5

10. Consider the sequence 5, 11, 17, 23, 29, … An explicit formula for any term a n of this sequence is a n = 5 + 6(n – 1). If we wanted to write a series for this sequence, we would use the following notation: To find an explicit formula for the series: 5 + 11 + 17 + 23 we write

11. Use the explicit formula for the sequence to write a formula for each series using sigma notation. Recall: Yesterday, we learned the explicit formula for an arithmetic sequence: a n = a 1 + (n – 1)d (7). 5 + 9 + 13 + 17 + 21 = (8). 12 + 9 + 6 + 3 + 0 + (– 3) + (– 6) + (– 9) = i=1 5 [5 + (i – 1)4] i=1 8 [12 + (i – 1)(-3)]

12. Check for Understanding: (9). Find S 3 for the sequence 7, 10, 13, 15,... (10). Find the sum (11). Write the sigma notation for the series: -2 + 3 + 8 + 13 + 18 + 23 = S 3 = 7 + 10 + 13 = 30 = 6 + 7 + 8 = 21 i=1 6 [-2 + (i – 1)5]

13. Investigation 2: For an arithmetic series with n terms, the sum of the first n terms is Note: Remember that a n = a 1 + (n – 1)d.

15. What type of equation is contained in the box?______________ Any sequence of partial sums of an arithmetic sequence is an example of a quadratic function because n is always raised to the second power. This formula allows us to find a sum without identifying each term in the series!!! Quadratic

16. Calculate each sum using the formula for the sum of an arithmetic series: (12). = a 1 = 2 + 5 = 7 a n = a 8 = 16 + 5 = 21 n = 8

17. (13). a 1 = 1 – 3 = -2 a n = a 10 = 10 – 3 = 7 n = 10

18. (14). a 1 = 3/4 a n = a 15 = (3/4)(15) = 45/4 n = 15

19. Write the summation notation for each series, then find the sum: (15). 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 a 1 = 1 a n = a 10 = 19 n = 10

20. Write the summation notation for each series, then find the sum: (16). 12 + 9 + 6 + 3 + 0 + (– 3) + (– 6) + (– 9) a 1 = 12 a n = a 8 = -9 n = 8

21. Write the summation notation for each series, then find the sum: (17). 5 + 9 + 13 +17 + 21 +... + 49 + 53 How many terms are there in the summation? a n = 5 + (n – 1)4 53 = 5 + 4n – 4 53 = 4n + 1 52 = 4n 13 = n

22. Write the summation notation for each series, then find the sum: (17). 5 + 9 + 13 +17 + 21 +... + 49 + 53 a 1 = 5 a n = a 13 = 53 n = 13

23. Write the summation notation for each series, then find the sum: (18). 1 + 5 + 9 +13 + 17 +... (Find S 50 ) a 1 = 1 a n = a 50 = 197 n = 50