Geomtric Series

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The midpoints of the sides of a series of a square of side 1 are joined to form a new sqaure. This procedure is repeated for each new square.
a) find the sum of the areas all the squares

b) find the sum of the perimeters of all the aquares

  • Geomtric Series -

    the area of the first square is 1
    clearly each successive square is 1/2 of the previous one.
    check: side of second square:
    (1/2)^2 + (1/2)^2 = x^
    x^2 = 1/2
    x = 1/√2
    so the area = (1/√2)^2 = 1/2

    so we have an infinite series
    1 + 1/2 + 1/4 + ...
    a=1, r=1/2
    S = 1/(1-r) = 1/(1-1/2) = 2

    hint for b)
    side of first square = 1
    side of second square = 1/√2
    ....

    take it from there.

  • Geomtric Series -

    so the perimeter is four and the series stay the same?

  • Geomtric Series -

    Huh?
    the perimeter of the first square =4
    the perimeter of the second square = 4*(1/√2) = 4/√2

    so a=4, r = 1/√2

    sum = 4/(1 - 1/√2)
    =

    you finish it , ok?

  • Geomtric Series -

    yeah that is what I was asking...thanx.!

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