The ancient Babylonians (circa 1700 B.C.) approximated √N by applying the formula

Show that this formula can be derived from the formula for Newton’s method in Exercise 33, and then use it to estimate √1,265. Repeat the formula until two consecutive approximations agree to four decimal places. Use your calculator to check your result.

Data from Exercises 33

Show that when Newton’s method is applied repeatedly, the nth approximation is obtained from the (n − 1)st approximation by the formula

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Xn+1 = 1/(x₁ N Xxn + Xn. for n = 1, 2, 3, ...