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A sequence of four integers is both arithmetic and geometric. if the sum of the four numbers is 20, find the first integer in the sequence.

  • Math -

    5. The only sequences that are both
    arithmetic and geometric are constant
    sequences, in which all the terms must
    be the same. For us to have a sum of 20,
    each term of the sequence must be 5. We
    can show this algebraically by solving
    the system of equations a + d = ar and
    a + 2d = ar² where a represents the first term of the sequence, d the common
    difference of the arithmetic sequence,
    and r the common ratio of the geometric sequence. This system produces the
    solution d = 0 and r = 1.

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