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Arithmetic Sequences & Series - nth Term Calculation

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Determine the 10th term of an arithmetic sequence using the nth term formula. Find missing terms and means in the sequence. Write equations for nth terms.

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Arithmetic Sequences & Series - nth Term Calculation

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  1. 11.2: Arithmetic Sequences & Series

  2. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below.

  3. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence.

  4. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d

  5. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = a1 + (10 – 1)d

  6. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = 55 + (10 – 1)d

  7. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = 55 + (10 – 1)(-6)

  8. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = 55 + (10 – 1)(-6)

  9. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = 55 + (10 – 1)(-6) a10 = 55 + (9)(-6)

  10. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = 55 + (10 – 1)(-6) a10 = 55 + (9)(-6) a10 = 55 – 54

  11. nth Term of an Arithmetic Sequence: an = a1 + (n – 1)d Ex. 1Determine the following using the table below. a) Find the 10th term in the sequence. an = a1 + (n – 1)d a10 = 55 + (10 – 1)(-6) a10 = 55 + (9)(-6) a10 = 55 – 54 a10 = 1

  12. b) Write an equation for the nth term of the sequence.

  13. b) Write an equation for the nth term of the sequence. an = a1 + (n – 1)d

  14. b) Write an equation for the nth term of the sequence. an = a1 + (n – 1)d an = 55 + (n – 1)(-6)

  15. b) Write an equation for the nth term of the sequence. an = a1 + (n – 1)d an = 55 + (n – 1)(-6) an = 55 - 6(n – 1)

  16. b) Write an equation for the nth term of the sequence. an = a1 + (n – 1)d an = 55 + (n – 1)(-6) an = 55 - 6(n – 1) an = 55 - 6n + 6

  17. b) Write an equation for the nth term of the sequence. an = a1 + (n – 1)d an = 55 + (n – 1)(-6) an = 55 - 6(n – 1) an = 55 - 6n + 6 an = - 6n + 61

  18. b) Write an equation for the nth term of the sequence. an = a1 + (n – 1)d an = 55 + (n – 1)(-6) an = 55 - 6(n – 1) an = 55 - 6n + 6 an = - 6n + 61

  19. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1

  20. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 ***Find the missing terms in the sequence!

  21. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6

  22. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6

  23. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1

  24. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 an = a1 + (n – 1)d

  25. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 an = a1 + (n – 1)d -1 = 24+ (6 – 1)d

  26. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 an = a1 + (n – 1)d -1 = 24+ (6 – 1)d -1 = 24 + 5d -25 = 5d -5 = d

  27. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 an = a1 + (n – 1)d -1 = 24+ (6 – 1)d -5 = d a1 = 24

  28. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 an = a1 + (n – 1)d -1 = 24+ (6 – 1)d -5 = d a1 = 24 a2 = 24 + (-5) = 19

  29. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 an = a1 + (n – 1)d -1 = 24+ (6 – 1)d -5 = d a1 = 24 a2 = 24 + (-5) = 19 a3 = 19 + (-5) = 14

  30. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 d = -5 a1 = 24 a2 = 24 + (-5) = 19 a3 = 19 + (-5) = 14 a4 = 14 + (-5) = 9

  31. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 d = -5 a1 = 24 a2 = 24 + (-5) = 19 a3 = 19 + (-5) = 14 a4 = 14 + (-5) = 9 a5 = 9 + (-5) = 4

  32. Ex. 2 Find the arithmetic means in the sequence below. 24, ___, ___, ___, ___, -1 a1a2a3a4a5a6 n = 6 a1 = 24 a6 = -1 d = -5 a1 = 24 a2 = 24 + (-5) = 19 a3 = 19 + (-5) = 14 a4 = 14 + (-5) = 9 a5 = 9 + (-5) = 4

  33. Sum of an Arithmetic Series

  34. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following:

  35. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ]

  36. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ]

  37. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d] OR Sn = ½n[ a1 + an ]

  38. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ]

  39. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following:

  40. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26

  41. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26

  42. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26 Sn = ½n[ a1 + an ]

  43. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26 Sn = ½n[ a1 + an ] Sn = ½(26)[ 58- 7 ]

  44. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26 Sn = ½n[ a1 + an ] Sn = ½(26)[ 58 - 7 ] Sn = ½(26)[ 51 ]

  45. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26 Sn = ½n[ a1 + an ] Sn = ½(26)[ 58 - 7 ] Sn = ½(26)[ 51 ] Sn = 13[ 51 ]

  46. Sum of an Arithmetic Series The sum Snof the first n terms of an arithmetic series is given by the following: Sn = ½n[ 2a1 + (n – 1)d ] OR Sn = ½n[ a1 + an ] Ex. 3 Find Snfor each of the following: a) a1= 58, an = -7, n = 26 Sn = ½n[ a1 + an ] Sn = ½(26)[ 58 - 7 ] Sn = ½(26)[ 51 ] Sn = 13(51) = 663

  47. Ex. 416 ∑ (4k – 2) k = 1

  48. Ex. 416 ∑ (4k – 2) k = 1 n = 16

  49. Ex. 416 ∑ (4k – 2) k = 1 n = 16 a1= 4(1) – 2 = 2

  50. Ex. 416 ∑ (4k – 2) k = 1 n = 16 a1= 4(1) – 2 = 2 an= 4(16) – 2 = 62

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