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Math Help - Geometric Sequences

  1. #1
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    Geometric Sequences

    In a square, one can connect the midpoints of adjacent sides to form second square. One can then connect the midpoints of adjacent sides of the second square to form a third square and continue in the same manner to form more squares successively.

    Suppose the area of the initial square is 1 square unit, and consider the sequence of areas of the squares formed in this way.

    Do these area form a geometric sequence? If they do, what are the first term and common ratio of the geometric sequence? If they do not, what sort of sequence do they form?

    Please Help!!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Geometric Sequences

    For any two successive squares in the sequence, what is the ratio of the smaller to the larger?
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  3. #3
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    Re: Geometric Sequences

    Hello, skweres1!

    Did you make a sketch?


    In a square, one can connect the midpoints of adjacent sides to form second square.
    One can then connect the midpoints of adjacent sides of the second square to form a third square
    and continue in the same manner to form more squares successively.

    Suppose the area of the initial square is 1 square unit,
    and consider the sequence of areas of the squares formed in this way.

    Do these areas form a geometric sequence? . Yes!
    If they do, what are the first term and common ratio of the geometric sequence?
    If they do not, what sort of sequence do they form?
    Code:
          *------.*.------*
          |    .*:|:*.    |
          |  .*:::|:::*.  |
          |.*:::::|:::::*.|
          *:-:-:-:+:-:-:-:*
          | *:::::|:::::* |
          |   *:::|:::*   |
          |     *:|:*     |
          *-------*-------*
    We see that the inner square's area is one-half the area of the outer square.

    The areas form a geometric sequence:
    . . . first term a = 1, common ratio r = \tfrac{1}{2}.
    Thanks from skweres1
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