A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)is
Side of first square = 8cm.
Side of second square made by joining mid-points of first square = $$\frac{8}{\sqrt2}$$
Similarly side of third square = $$\frac{8}{\sqrt2\times\sqrt2}$$ and so on.
Now summation of areas will be $$8^2+(\frac{8}{\sqrt2})^2+(\frac{8}{\sqrt2\times\sqrt2})^2$$ .........
or $$8^2 ( 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}.....)$$
or $$64 \times (\frac{1}{1-\frac{1}{2}})$$ (As we know sum of an infinite G.P. is $$\frac{a}{1-r}$$ where a is first term and r is common ratio)
or 128 sq. cm.
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