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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.

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Sequence and Series Review Problems

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  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

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