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Question
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
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Solution
Area of the 1st square = 12
Area of the 2nd square = `(1/sqrt2)^2`
Area of the 3rd square = `(1/2)^2`
and so on
∴ Sum of the areas of all the squares
= `1^2+(1/sqrt2)^2+(1/2)^2+...`
= `1+1/2+1/4+...`
∴ a = 1, r = `1/2`
Since, |r| = `|1/2|<1`
∴ sum to infinity exists.
∴ Sum of the areas of all the squares = `1/(1-1/2)` = 2
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