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This exercise is based on an ancient Babylonian method for approximating the square root of a positive number c. Assume that c lies in the interval (1,4) for simplicity. Set x0=3 / 2 . For n ≥0, define xn 1=(1)/(2)(xn (c)/(xn)) a. Use the definition of xn 1 to obtain the identity |xn 1-√(c)|=(xn-√(c))^2 / 2 xn b. Use part (a) to deduce that if xn approximates √(c) to k decimal places, then xn 1 approximates √(c) to 2 k decimal places. c. Use this algorithm to compute √(3.75) to 8 decimal places.
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