1. Let $$S_{1}, S_{2},...$$ be the squares such that for each n ≥ 1, the length of the diagonal of $$S_{n}$$ is equal to the length of the side of $$S_{n}+1$$. If the length of the side of $$S_{3}$$ is 4 cm, what is the length of the side of $$S_{n}$$ ?
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By: anil on 05 May 2019 02.38 am
Length of side of $$S_{n + 1}$$ = Length of diagonal of $$S_n$$ => Length of side of $$S_{n + 1}$$ = $$sqrt{2}$$ (Length of side of $$S_{n}$$) => $$frac{ extrm{Length of side of }S_{n + 1}}{ extrm{Length of side of }S_n} = sqrt{2}$$ => Sides of $$S_1 , S_2 , S_3 , S_4,........, S_n$$ form a G.P. with common ratio, $$r = sqrt{2}$$ It is given that, $$S_3 = ar^2 = 4$$ => $$a (sqrt{2})^2 = 4$$ => $$a = frac{4}{2} = 2$$ $$ herefore$$ $$n^{th}$$ term of G.P. = $$a (r^{n - 1})$$ = $$2 (sqrt{2})^{n - 1}$$
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