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27 March, 18:59

In an arithmetic sequence, a17 = - 40 and

a28 = - 73. Explain how to use this information to write a recursive formula for this sequence

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Answers (2)
  1. 27 March, 19:08
    0
    Tn = 2Tn-1 - Tn-2

    Step-by-step explanation:

    Before we can generate the recursive sequence, we need to find the nth term of the given sequence.

    nth term of an AP is given as:

    Tn = a + (n-1) d

    If a17 = - 40

    T17 = a + (17-1) d = - 40

    a+16d = - 40 ... (1)

    If a28 = - 73

    T28 = a + (28-1) d = - 73

    a+27d = - 73 ... (2)

    Solving both equations simultaneously using elimination method.

    Subtracting 1 from 2 we have:

    27d - 16d = - 73 - (-40)

    11d = - 73+40

    11d = - 33

    d = - 3

    Substituting d = - 3 into 1

    a+16 (-3) = - 40

    a - 48 = - 40

    a = - 40+48

    a = 8

    Given a = 8, d = - 3, the nth term of the sequence will be

    Tn = 8 + (n-1) (-3)

    Tn = 8 + (-3n+3)

    Tn = 8-3n+3

    Tn = 11-3n

    Given Tn = 11-3n and d = - 3

    Tn-1 = Tn - d ... (3)

    Tn-1 = 11-3n + 3

    Tn-1 = 14-3n

    Tn-2 = Tn-2d ... (4)

    Tn-2 = 11-3n-2 (-3)

    Tn-2 = 11-3n+6

    Tn-2 = 17-3n

    From 3, d = Tn - Tn-1

    From 4, d = (Tn - Tn-2) / 2

    Equating both common difference

    (Tn - Tn-2) / 2 = Tn - Tn-1

    Tn - Tn-2 = 2 (Tn - Tn-1)

    Tn - Tn-2 = 2Tn-2Tn-1

    2Tn-Tn = 2Tn-1 - Tn-2

    Tn = 2Tn-1 - Tn-2

    The recursive formula will be

    Tn = 2Tn-1 - Tn-2
  2. 27 March, 19:12
    0
    The difference between the given terms is

    -73 - (-40) = - 33.

    The difference between the term numbers is 28 - 17 = 11.

    Dividing - 33 / 11 = - 3.

    The common difference is - 3.

    The recursive formula is the previous term minus 3, or an = an - 1 - 3 where a17 = - 40.
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