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Math Help - Arithmetic Sequences

  1. #1
    Junior Member Godfather's Avatar
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    Arithmetic Sequences

    For a given arithmetic sequence the sum of the first fifty terms is 200 and the sum of the next fifty term is 2700. What is the first term of the sequence?
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  2. #2
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    Hello, Godfather!

    For a given arithmetic sequence the sum of the first fifty terms is 200
    . . and the sum of the next fifty term is 2700.
    What is the first term of the sequence?

    The sum of the first n terms of an arithmetic sequence is given by:
    . . S(n) .= .(n/2)[2a + (n-1)d]

    The sum of the first 50 terms is 200.
    . . S(50) .= .(50/2)[2a + 49d] .= .200 . . 2a + 49d .= .8 .[1]

    The sum of the next 50: (sum of the first 100) - (sum of the first 50)
    . . (100/2)[2a + 99d] - 200

    Hence, we have: .50[2a + 99d] - 200 .= .2700 . . 2a + 99d .= .58 .[2]


    Subtract [1] from [2]: .50d .= .50 . . d = 1

    Substitute into [1]: .2a + 49(1) .= .8 . . a .= .-41/2

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