1 / 50

Sequence and Series Review Problems

Sequence and Series Review Problems. Sequence Review Problems. Example 1: Find the explicit formula for the following arithmetic sequence: The explicit formula for arithmetic sequences is: You must replace the a 1 and d. 1, 6, 11, 16, …. a n = a 1 + (n – 1) d. Sequence Review Problems.

oma
Download Presentation

Sequence and Series Review Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Sequence and Series Review Problems

  2. Sequence Review Problems • Example 1: Find the explicit formula for the following arithmetic sequence: • The explicit formula for arithmetic sequences is: • You must replace the a1 and d 1, 6, 11, 16, … an = a1 + (n – 1)d

  3. Sequence Review Problems • First you must find the common difference between the numbers: 1, 6, 11, 16, … 5 5 d =

  4. Sequence Review Problems • Then find a1 a1 1, 6, 11, 16, …

  5. Sequence Review Problems • Plug them both into the equation: • Distribute the d • Combine like terms a1 = 1 an = 1 + (n – 1)5 d = 5 an = 1 + 5n – 5 an = 5n – 4

  6. Practice 1 • Put the following sequences into explicit form: • 1, 3, 5, 7, … • -2, 1, 4, 7, … • 9, 6, 3, 0, … • -67, -60, -53, …

  7. Sequence Review Problems • Example 2: Find the recursive formula for the following arithmetic sequence: • The recursive formula for an arithmetic sequence is: • You need to find the difference a1 = 3, a5 = 15 an = an-1 + d

  8. Sequence Review Problems • The difference formula: • an = the second or last number in the sequence. It can be any number after the first number • a1 represents the first number you’re given (if not a1, then use the earliest term)

  9. Sequence Review Problems • In this case we have a1 and a5, so a1 will be the first term and a5 will be the last a1 = 3, a5 = 15

  10. Sequence Review Problems • Plug d into the equation: an = an-1 + d an = an-1 + 3

  11. Practice 2 • Find the recursive formula for the following arithmetic sequences • a1 = 2, a6 = 12 • a1 = 17, a8 = -25 • a3 = 5, a10 = 54 • a7 = 23, a19 = -85

  12. Sequence Review Problems • Example 3: Find the explicit formula for the following geometric sequence: • The explicit formula for a geometric sequence is: • We must replace the a1 and r 1, 2, 4, 8, …

  13. Sequence Review Problems • r is the common ration determined by dividing the second from the first number: 1, 2, 4, 8, …

  14. Sequence Review Problems • And you know a1 is the first term in the series a1 1, 2, 4, 8, …

  15. Sequence Review Problems • Plug them both into the equation: • DO NOT combine the numbers!!! • In this case 1 times a number is just that numbers so you don’t need the 1 a1 = 1 r = 2

  16. Practice 3 • Find the explicit formula for the following geometric sequences • 1, 3, 9, … • 3, 12, 48, … • 2, -10, 50, … • 36, 18, 9, …

  17. Sequence Review Problems • Example 4: Find the recursive formula for the following geometric sequence: • The recursive formula for a geometric sequence is 27, 9, 3, … an = r*an-1 times

  18. Sequence Review Problems • Find r by dividing the second term by the first 27, 9, 3, …

  19. Sequence Review Problems • Plug r into the equation an = r*an-1

  20. Practice 4 • Find the recursive formulas for the following geometric sequences • 6, 12, 24, … • 180, 60, 20, … • 8, -32, 128, … • -1, -2, -4, …

  21. Series Review Problems • Example 5: Find the sum of the arithmetic series • This is an arithmetic series (any time you see the word SUM you’re using series) • Since you have the first and last term, use the formula a1 = 3, a6 = 19

  22. Series Review Problems The sum of the first n terms The number of terms you’re adding

  23. Series Review Problems The last term you’re adding The first term you’re adding

  24. Series Review Problems • Since you have the first and the 6th term, there are a total of 6 terms you’re adding so a1 = 3, a6 = 19 n = 6 • Plug everything into the equation

  25. Series Review Problems * • Simplify • That is the sum of the first six numbers in the series

  26. Series Review Problems • Example 6: Find the sum of the following series: • In this case we don’t have a1, so we’ll use a5as the starting point a5 = 5, a30 = 78

  27. Series Review Problems • We need to find the number of terms we’re adding up • We are starting with the 5th term, so we are leaving out terms 1, 2, 3, and 4 • That means we are adding 30 – 4 terms • So: n = 26

  28. Series Review Problems • In the formula, use a5 for the first number and a30 for the last • Use your calculator a5 = 5, a30 = 78 n = 26

  29. Practice 5 • Find the sum of the following arithmetic series • a1 = 2, a6 = 12 • a1 = 17, a8 = -25 • a3 = 5, a10 = 54 • a7 = 23, a19 = -85

  30. Series Review Problems • Example 7: Find the 40th partial sum of the following arithmetic series: • We aren’t given the last term in the series, so we can use the other arithmetic series formula: 1 + 4 + 7 + …

  31. Series Review Problems • 40th partial sum means add the first 40 terms so: • We can also find d: n = 40 1 + 4 + 7 + … 3 3 d =

  32. Series Review Problems • Plug everything into your formula

  33. Practice 6 • Find the 20th partial sum of the series 1 + 5 + 9 + … • Find the 50th partial sum of the series 2 + -2 + -6, … • Find the 100th partial sum of the series 3 + 7 + 11 + …

  34. Series Review Problems • Example 8: Find the sum of the geometric series: • Use the geometric series formula: a1 = 2, r = 3, n = 6

  35. Series Review Problems • We have everything we need to plug in and solve the problem a1 = 2 r = 3 n = 6

  36. Practice 7 • Find the sum of the following geometric series • a1 = 2, r = 4, n = 5 • a1 = 3, r = -2, n = 10 • a1 = 12, r = ½ , n = 6

  37. Series Review Problems • Put the following series in sigma notation: • There are 5 terms so the bottom is n = 1 and the top is 5 1 + 4 + 7 + 11 + 15 a1 a5

  38. Series Review Problems • We now need an explicit formula to go in front • The series is arithmetic since there is a common difference ? 1 + 4 + 7 + 10 + 13 3 3 d =

  39. Series Review Problems • Use the arithmetic explicit formula: an = a1 + (n – 1)d an = 1+ (n – 1)3 an = 1+ 3n – 3 an = 3n – 2

  40. Series Review Problems • Plug your formula into sigma 3n – 2

  41. Practice 8 • Put the following series into sigma notation • 1 + 3 + 5 + 7 + 9 + 11 + 13 • 1 + 2 + 4 + 8 + 16 + 32 • 1 + -3 + -7 + -11

  42. Series Review Problems • Solve: n = 6 Start at the first term Add to the sixth term

  43. Series Review Problems • Since the formula is arithmetic (the n is not an exponent) we’ll use the formula: • We need to find the first and last term (6th)

  44. Series Review Problems • To find the first term, plug 1 into the equation • Now plug it all into your formula a1 =3(1) + 2 = 5 a6 =3(6) + 2 = 20 n = 6

  45. Series Review Problems • Solve

  46. Series Review Problems • Solve: Start at the 3rd term Add to the 8th term

  47. Series Review Problems • This is a geometric series since n is an exponent • Use the formula: • We need a1, r, and n

  48. Series Review Problems • Plug in 3 to find the first term we’re adding • r is the number being raised to the power • n is the number we are adding up: 8 – 2 = 6 (remember, we’re leaving out the first 2 numbers)

  49. Series Review Problems • Plug everything into the formula a3 = 4 r = 2 n = 6

  50. Practice 9 • Solve the following

More Related