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To find the n^th term and the three terms of the arithmetic sequence, we first need to determine the common difference (d) and the first term (a₁).

Mathematics

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Answer

121 - 7n\text{121 - 7n}

114, 107, 100\text{114, 107, 100}

Step-by-step solution

4 steps

To find the nthn^{th} term and the three terms of the arithmetic sequence, we first need to determine the common difference (dd) and the first term (a1a_1).

Given:
a4=93a_4 = 93
a8=65a_8 = 65

  1. Find the common difference (dd).

    In an arithmetic sequence, the nthn^{th} term can be expressed as an=ak+(n−k)da_n = a_k + (n-k)d.
    Using a8a_8 and a4a_4:

    a8=a4+(8−4)da_8 = a_4 + (8-4)d a8=a4+4da_8 = a_4 + 4d

    Substitute the given values:

    65=93+4d65 = 93 + 4d

    Subtract 93 from both sides:

    65−93=4d65 - 93 = 4d −28=4d-28 = 4d

    Divide by 4:

    d=−284d = \frac{-28}{4} d=−7d = -7
  2. Find the first term (a1a_1).

    The general formula for the nthn^{th} term of an arithmetic sequence is an=a1+(n−1)da_n = a_1 + (n-1)d.
    Using a4=93a_4 = 93 and d=−7d = -7:

    a4=a1+(4−1)da_4 = a_1 + (4-1)d 93=a1+3(−7)93 = a_1 + 3(-7) 93=a1−2193 = a_1 - 21

    Add 21 to both sides:

    a1=93+21a_1 = 93 + 21 a1=114a_1 = 114
  3. Find the nthn^{th} term (ana_n).

    Using the general formula an=a1+(n−1)da_n = a_1 + (n-1)d with a1=114a_1 = 114 and d=−7d = -7:

    an=114+(n−1)(−7)a_n = 114 + (n-1)(-7) an=114−7n+7a_n = 114 - 7n + 7 an=121−7na_n = 121 - 7n
  4. Find the first three terms of the sequence.

    The first term is a1=114a_1 = 114.
    The second term is a2=a1+d=114+(−7)=107a_2 = a_1 + d = 114 + (-7) = 107.
    The third term is a3=a2+d=107+(−7)=100a_3 = a_2 + d = 107 + (-7) = 100.

    The nthn^{th} term of the arithmetic sequence is 121 - 7n\boxed{\text{121 - 7n}}.
    The first three terms of the sequence are 114, 107, 100\boxed{\text{114, 107, 100}}.

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