Homework Help: Geomtric Series

Posted by Janet on Wednesday, March 18, 2009 at 7:53pm.

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 - Reiny, Wednesday, March 18, 2009 at 8:25pm
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 - Janet, Wednesday, March 18, 2009 at 8:34pm
so the perimeter is four and the series stay the same?
Geomtric Series - Reiny, Wednesday, March 18, 2009 at 8:43pm
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 - Janet, Wednesday, March 18, 2009 at 8:53pm
yeah that is what I was asking...thanx.!

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