Let c > 0. A sequence is given by Use the method of mathematical induction to pr

Question

Let c > 0. A sequence is given by Use the method of mathematical induction to prove that (xn) is an increasing sequence. Use the method of mathematical induction to prove that Xn

Explanation / Answer

a) For n = 1, x1 = sqrt(c) x2 = sqrt(c + sqrt(c)) sqrt(c) > 0, so c + sqrt(c) > c. As sqrt is a monotone transformation, then sqrt(c + sqrt(c)) > sqrt(c), and x2 > x1 Assume up to n=k Then, for n = k+1 x k+1 = sqrt(c + xk) xk+2 = sqrt(c + xk+1) Yet, by assumption for n=k, xk+1 > xk. Thus, c + xk+1 > c + xk, and sqrt(c+xk+1) > sqrt(c + xk) , so xk+2 > xk+1 b) For n = 1, x1 = sqrt(c)

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