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Squares Diagonal and Side Length Progression

XAT 2010 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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Let S1,S2,...S_1, S_2, ... be the squares such that for each n≥1n \ge 1, the length of the diagonal of SnS_n is equal to the length of the side of Sn+1S_{n+1}. If the length of the side of S3S_3 is 4 cm, what is the length of the side of SnS_n?

  1. A.

    22n+122^{\frac{2n+1}{2}}

  2. B.

    2⋅(n−1)2 \cdot (n - 1)

  3. C.

    2n−12^{n-1}

  4. D.

    2n+122^{\frac{n+1}{2}}

  5. E.

    None of these

Answer

D

Explanation

Diagonal of Sn=2⋅side(Sn)=side(Sn+1)S_n = \sqrt{2} \cdot \text{side}(S_n) = \text{side}(S_{n+1}). Thus, the sides form a GP with common ratio r=2r = \sqrt{2}. Given side(S3)=ar2=a(2)2=2a=4  ⟹  a=2\text{side}(S_3) = a r^2 = a (\sqrt{2})^2 = 2a = 4 \implies a = 2. side(Sn)=arn−1=2(2)n−1=21⋅2n−12=2n+12\text{side}(S_n) = a r^{n-1} = 2 (\sqrt{2})^{n-1} = 2^1 \cdot 2^{\frac{n-1}{2}} = 2^{\frac{n+1}{2}}.

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