algebra

the third term of an arithmetic sequence is 14 and the ninth term is -1. Find the first four terms of the sequence

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  1. ninth -1, third 14
    it goes now by 15 in six steps; step 15/6 or 2.5
    third through tenth terms
    14, 11.5, 9, 6.5, 4, 1.5, -1, -3.5,...

    check that

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  2. third term is 14 and ninth term is -1

    14 - 2.5 = 11.5 > 4th term
    11.5 - 2.5 = 9 > 5th term
    9 - 2.5 = 6.5 > 6th term
    6.5 - 2.5 = 4 > 7th term
    4 - 2.5 = 1.5 > 8th term
    1.5 - 2.5 = -1 > 9th term

    The terms are decreasing by 2.5 > (-2.5)
    To go from one term to the next, subtract 2.5.

    The common difference is -2.5
    d = -2.5

    Arithmetic Sequence Formula:
    Tn = Tn + d(n-1)

    a = 1st term
    n = nth term
    d = common difference

    We don't know the first term yet!

    Tn = T1 + d(n-1)

    Substitute 3 for n, and -2.5 for d

    T3 = T1 + -2.5(3-1)

    Now substitute 14 for T3

    14 = T1 + -2.5(2)
    14 = T1 -5
    14 + 5 = T1 -5 +5
    19 = T1

    T1 = 19
    T2 = 16.5
    T3 = 14
    T4 = 11.5
    Now to find a term:

    Tn = T1 + d(n-1)

    T2 = 19 + -2.5(2-1) T3 = 19 + -2.5(3-1)
    T2 = 19 + -2.5(1) T3 = 19 + -2.5(2)
    T2 = 19 + -2.5 T3 = 19 + -5
    T2 = 16.5 T3 = 14
    __________________________

    You can also use this formula:

    Tn = dn + 21.5

    T1 = -2.5(1) + 21.5
    T1 = 19

    T4 = -2.5(4) + 21.5
    T4 = 11.5

    T9 = -2.5(9) + 21.5
    T9 = -1

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