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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 perimeters of all the squares
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Solution
Perimeter of 1st square = 4
Perimeter of 2nd square = `4(1/sqrt(2))`
Perimeter of 3rd square = `4(1/2)`
and so on.
∴ Sum of the perimeters of all the squares
= `4 + 4(1/sqrt(2)) + 4(1/2) + ...`
= `4(1 + 1/sqrt(2) + (1/sqrt(2))^2 + ...)`
The terms `1,1/sqrt(2), (1/sqrt(2))^2, ...` are in G.P.
∴ a = 1, r = `1/sqrt(2)`
Since, |r| = `|1/sqrt(2)| < 1`
∴ sum to infinity exists.
∴ Sum of the perimeters of all the squares
= `4(1/(1 - 1/sqrt(2)))`
= `(4sqrt(2))/(sqrt(2) - 1)`
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