11.2: Arithmetic Sequences & Series

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11.2: Arithmetic Sequences & Series. n th Term of an Arithmetic Sequence: a n = a 1 + ( n – 1) d Ex. 1 Determine the following using the table below. n th Term of an Arithmetic Sequence: a n = a 1 + ( n – 1) d Ex. 1 Determine the following using the table below. - PowerPoint PPT Presentation

Text of 11.2: Arithmetic Sequences & Series

  • 11.2: Arithmetic Sequences & Series

  • nth Term of an Arithmetic Sequence:an = a1 + (n 1)d

    Ex. 1Determine the following using the table below.

    55494337a1a2a3a4

  • 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.

    55494337a1a2a3a4

  • 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

    55494337a1a2a3a4

  • 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

    55494337a1a2a3a4

  • 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

    55494337a1a2a3a4

  • 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)

    55494337a1a2a3a4

  • 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)

    55494337a1a2a3a4

  • 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)

    55494337a1a2a3a4

  • 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

    55494337a1a2a3a4

  • 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

    55494337a1a2a3a4

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

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

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

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

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

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

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

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

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

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6

    n = 6a1 = 24a6 = -1

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6

    n = 6a1 = 24a6 = -1 an = a1 + (n 1)d

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6

    n = 6a1 = 24a6 = -1 an = a1 + (n 1)d -1 = 24 + (6 1)d

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1 an = a1 + (n 1)d -1 = 24 + (6 1)d -1 = 24 + 5d -25 = 5d -5 = d

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1 an = a1 + (n 1)d -1 = 24 + (6 1)d -5 = da1 = 24

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1 an = a1 + (n 1)d -1 = 24 + (6 1)d -5 = da1 = 24a2 = 24 + (-5) = 19

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1 an = a1 + (n 1)d -1 = 24 + (6 1)d -5 = da1 = 24a2 = 24 + (-5) = 19a3 = 19 + (-5) = 14

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1d = -5a1 = 24a2 = 24 + (-5) = 19a3 = 19 + (-5) = 14a4 = 14 + (-5) = 9

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1d = -5a1 = 24a2 = 24 + (-5) = 19a3 = 19 + (-5) = 14a4 = 14 + (-5) = 9a5 = 9 + (-5) = 4

  • Ex. 2 Find the arithmetic means in the sequence below.24, ___, ___, ___, ___, -1a1a2a3a4a5 a6n = 6a1 = 24a6 = -1d = -5a1 = 24a2 = 24 + (-5) = 19a3 = 19 + (-5) = 14a4 = 14 + (-5) = 9a5 = 9 + (-5) = 4

  • Sum of an Arithmetic Series

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following:

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26Sn = n[ a1 + an ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26Sn = n[ a1 + an ]Sn = (26)[ 58 - 7 ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26Sn = n[ a1 + an ]Sn = (26)[ 58 - 7 ]Sn = (26)[ 51 ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26Sn = n[ a1 + an ]Sn = (26)[ 58 - 7 ]Sn = (26)[ 51 ] Sn = 13[ 51 ]

  • Sum of an Arithmetic SeriesThe sum Sn of the first n terms of an arithmetic series is given by the following: Sn = n[ 2a1 + (n 1)d ]OR Sn = n[ a1 + an ]

    Ex. 3 Find Sn for each of the following:a) a1 = 58, an = -7, n = 26Sn = n[ a1 + an ]Sn = (26)[ 58 - 7 ]Sn = (26)[ 51 ] Sn = 13(51) = 663

  • Ex. 4 16 (4k 2) k = 1

  • Ex. 4 16 (4k 2) k = 1

    n = 16

  • Ex. 4 16 (4k 2) k = 1

    n = 16a1 = 4(1) 2 = 2

  • Ex. 4 16 (4k 2) k = 1

    n = 16a1 = 4(1) 2 = 2 an = 4(16) 2 = 62

  • Ex. 4 16 (4k 2) k = 1

    n = 16a1 = 4(1) 2 = 2 an = 4(16) 2 = 62

    Sn = n[ a1 + an ]

  • Ex. 4 16 (4k 2) k = 1

    n = 16a1 = 4(1) 2 = 2 an = 4(16) 2 = 62

    Sn = n[ a1 + an ]Sn = (16)[2 + 62]

  • Ex. 4 16 (4k 2) k = 1

    n = 16a1 = 4(1) 2 = 2 an = 4(16) 2 = 62

    Sn = n[ a1 + an ]Sn = (16)[2 + 62] Sn = 512