1. Arithmetic and Geometric Series (11.5) Short cuts

2. POD Consider a list of the first 100 positive integers. What sort of sequence is it? Add them. I’ll time how long it takes. What do we call a sum of terms in a sequence?

3. POD Consider a list of the first 100 positive integers. It’s an arithmetic sequence, with d = 1. We’re finding a series. We use sigma notation, or we can use S n notation. Want a short cut? It’s based off of something we did yesterday.

4. Arithmetic partial sums To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

5. Arithmetic partial sums To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = What do we do next? Can you think of a pattern?

6. Arithmetic partial sums To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 11+11+11+11+11+11+11+11+11+11=110 We added the first and last terms.

7. Arithmetic partial sums To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 11+11+11+11+11+11+11+11+11+11=110 We need only half of the number of sums here.

8. Arithmetic partial sums To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 11+11+11+11+11+11+11+11+11+11=110

9. Arithmetic partial sums Use this to answer the POD.

10. Arithmetic partial sums Use this to answer the POD. S 100 = 100/2(1 + 100) = 50(101) = 5050 This problem was posed to Karl Friedrich Gauss (1777-1855) in third grade, and he determined the pattern.

11. Arithmetic partial sums We have another tool, too. What is the formula for the n th term of an arithmetic sequence? How could we use it to change this formula?

12. Arithmetic partial sums We have another tool, too. What is the formula for the n th term of an arithmetic sequence? How could we use it?

13. Arithmetic partial sums We decide which tool to use based on the information given. What information would you need to either one?

14. Use them 1. Find the sum of the first 50 positive even integers. Which formula would you use? How does this compare to the sum of the first 100 positive integers?

15. Use them 1. Find the sum of the first 50 positive even integers. Which formula would you use? S 50 = 50/2(2 + 100) = 25(102) = 2550 How does this compare to the sum of the first 100 positive integers?

16. Use them 1. Find the sum of the first 50 positive even integers. Let’s look at it in summation notation.

17. Use them 2. Find S 12 for the series with t 1 = 3 and d = -4

18. Use them 2.Find S 12 for the series with t 1 = 3 and d = -4. S 12 = 12/2(2(3) + (11)(-4)) = 6(6-44) = 6(-38) = -228

19. Geometric shortcut The shortcut for a geometric series uses the first term and r.

20. Use it 3. Find S 8 for 1 + 2 + 4 +8 +….

21. Use it 3.Find S 8 for 1 + 2 + 4 +8 +….

22. Use it 4.Find S 20 for the series with first term of 11 and r = 1.3.

23. Use it 4.Find S 20 for the series with first term of 11 and r = 1.3.