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1105 ch 11 day 5

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• 1105 ch 11 day 5

1. 1. 11.1 Sequences & Summation Notation Day ThreeRevelation 3:20 "Here I am! I stand at the door and knock. Ifanyone hears my voice and opens the door, I will come in andeat with him, and he with me."
2. 2. The sum of the ﬁrst n terms of a sequence iscalled the nth partial sum and is denoted Sn
3. 3. The sum of the ﬁrst n terms of a sequence iscalled the nth partial sum and is denoted SnFind the indicated partial sums:1) S10 for − 3, − 6, − 9, − 12, K
4. 4. The sum of the ﬁrst n terms of a sequence iscalled the nth partial sum and is denoted SnFind the indicated partial sums:1) S10 for − 3, − 6, − 9, − 12, K−3 − 6 − 9 − 12 − 15 − 18 − 21− 24 − 27 − 30
5. 5. The sum of the ﬁrst n terms of a sequence iscalled the nth partial sum and is denoted SnFind the indicated partial sums:1) S10 for − 3, − 6, − 9, − 12, K−3 − 6 − 9 − 12 − 15 − 18 − 21− 24 − 27 − 30 −165
6. 6. Find the indicated partial sums:2) S6 for an = 10n − 6
7. 7. Find the indicated partial sums:2) S6 for an = 10n − 6 4 + 14 + 24 + 34 + 44 + 54
8. 8. Find the indicated partial sums:2) S6 for an = 10n − 6 4 + 14 + 24 + 34 + 44 + 54 174
9. 9. Find the indicated partial sums:2) S6 for an = 10n − 6 4 + 14 + 24 + 34 + 44 + 54 174A partial sum can be done on your calculatoras the sum of a sequence. Try it ... sum ( seq (10x − 6, x, 1, 6, 1))
10. 10. What is the syntax of this calculation? sum ( seq (10x − 6, x, 1, 6, 1))
11. 11. What is the syntax of this calculation? sum ( seq (10x − 6, x, 1, 6, 1)) explicit formulafor the sequence
12. 12. What is the syntax of this calculation? sum ( seq (10x − 6, x, 1, 6, 1)) explicit formulafor the sequence summation variable
13. 13. What is the syntax of this calculation? sum ( seq (10x − 6, x, 1, 6, 1)) explicit formulafor the sequence summation variable starts at
14. 14. What is the syntax of this calculation? sum ( seq (10x − 6, x, 1, 6, 1)) explicit formulafor the sequence summation variable starts at ends at
15. 15. What is the syntax of this calculation? sum ( seq (10x − 6, x, 1, 6, 1)) explicit formulafor the sequence increases by (called the step) summation variable starts at ends at
16. 16. Consider this: S1 , S2 , S3 , K Sn
17. 17. Consider this: S1 , S2 , S3 , K SnIt is a sequence of partial sums.More on this later this chapter ...
18. 18. Sigma Notation (or Summation Notation)
19. 19. Sigma Notation (or Summation Notation) n ∑a k = a1 + a2 + a3 +K + an k=1
20. 20. Sigma Notation (or Summation Notation) n ∑a k = a1 + a2 + a3 +K + an k=1 is read “the sum of ak as k goes from 1 to n “
21. 21. Sigma Notation (or Summation Notation) n ∑a k = a1 + a2 + a3 +K + an k=1 is read “the sum of ak as k goes from 1 to n “ k is the summation variable ... or ... index of summation
22. 22. Sigma Notation (or Summation Notation) n ∑a k = a1 + a2 + a3 +K + an k=1 is read “the sum of ak as k goes from 1 to n “ k is the summation variable ... or ... index of summation This is the math shorthand for doing the sum of a sequence just like what we did on the calculator!
23. 23. Example: Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26 using Sigma Notation
24. 24. Example: Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26 using Sigma NotationFind the explicit expression for this sequence.That will be your ak .Then, write the Sigma Notation!
25. 25. Example: Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26 using Sigma NotationFind the explicit expression for this sequence.That will be your ak .Then, write the Sigma Notation! 9 ∑ ( 3k − 1) k=1
26. 26. Example: Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26 using Sigma NotationFind the explicit expression for this sequence.That will be your ak .Then, write the Sigma Notation! 9 ∑ ( 3k − 1) k=1Practice calculating this on your calculator.
27. 27. Example: Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26 using Sigma NotationFind the explicit expression for this sequence.That will be your ak .Then, write the Sigma Notation! 9 ∑ ( 3k − 1) k=1Practice calculating this on your calculator. sum ( seq ( 3x − 1, x, 1, 9, 1)) 126
28. 28. Find the sum by hand and verify with calculator: 8 1) ∑ ( 3k − 4 ) k=1
29. 29. Find the sum by hand and verify with calculator: 8 1) ∑ ( 3k − 4 ) k=1 −1+ 2 + 5 + 8 + 11+ 14 + 17 + 20 76
30. 30. Find the sum by hand and verify with calculator: 8 1) ∑ ( 3k − 4 ) k=1 −1+ 2 + 5 + 8 + 11+ 14 + 17 + 20 76 sum ( seq ( 3x − 4, x, 1, 8, 1))
31. 31. Find the sum by hand and verify with calculator: 11 2) ∑4 k=7
32. 32. Find the sum by hand and verify with calculator: 11 2) ∑4 k=7 4+4+4+4+4 20
33. 33. Find the sum by hand and verify with calculator: 11 2) ∑4 k=7 4+4+4+4+4 20 sum ( seq ( 4, x, 7, 11, 1)) or sum ( seq ( 4, x, 1, 5, 1))
34. 34. Write this sum using Sigma Notation: 4 4 4 4 4 4 4 4 3) 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
35. 35. Write this sum using Sigma Notation: 4 4 4 4 4 4 4 4 3) 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 10 4 ∑k k=3
36. 36. Properties of Sums
37. 37. Properties of Sums n n n 1. ∑(a k + bk ) = ∑ ak + ∑ bk k=1 k=1 k=1
38. 38. Properties of Sums n n n 1. ∑(a k + bk ) = ∑ ak + ∑ bk k=1 k=1 k=1 n n n 2. ∑(a k − bk ) = ∑ ak − ∑ bk k=1 k=1 k=1
39. 39. Properties of Sums n n n 1. ∑(a k + bk ) = ∑ ak + ∑ bk k=1 k=1 k=1 n n n 2. ∑(a k − bk ) = ∑ ak − ∑ bk k=1 k=1 k=1 n n 3. ∑(c ⋅ a ) = c ⋅ ∑ a k k k=1 k=1
40. 40. HW #3“We must not only give what we have, we must alsogive what we are.” Desire Joseph Cardinal Mercier