Further mathematics

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An exponential sequence of positive terms and a linear sequence have the same first term.the sum of their first terms is 3 the sum of their second terms is 3/2,and the sum of their third terms is 6.find their fifth terms.

  • Further mathematics -

    Since the two sequences have the same first term, and they sum to 3, they both start with 3/2.

    Let the arithmetic sequence be 3/2, 3/2 + d, 3/2 + 2d, ...

    Let the geometric sequence be 3/2, 3/2 r, 3/2 r^2, ...

    3/2 + d + 3/2 r = 3/2
    so, d = -3/2 r

    3/2 + 2d + 3/2 r^2 = 6
    r = 3 or -1
    d = -9/2 or 3/2

    AS: 3/2, -3, -15/2, ...
    GS: 3/2, 9/2, 27/2, ...

    or

    AS: 3/2, 3, 9/2, ...
    GS: 3/2, -3/2, 3/2, ...

    I assume you can make it to the 5th terms of each sequence.

    Cool problem!

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