There are n concentric squares. The area of the innermost square is 1 unit and the distance between corresponding corners of any two consecutive squares is 1 unit. Consider the following statements:<br>I. The diagonal of the nth square is 2n+\sqrt{2}-2<br>II. The area included between nth square and (n-1)th square is independent of n<br>Which of the statements given above is/are correct?

Explanation

The innermost square has area 1, so its diagonal d_1 = \sqrt{1^2+1^2} = \sqrt{2}. For each subsequent square, the corners are 1 unit further along the diagonal, so the diagonal length increases by 2. Thus, d_n = \sqrt{2} + 2(n-1) = 2n + \sqrt{2} - 2 (Statement I is true). Area of the n-th square is \frac{d_n^2}{2}. The difference in area between consecutive squares grows with n, so it is not independent of n (Statement II is false).

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