Approaching One From Below

★ 8/10Monotonicity sequences Estimation Telescoping inequality

Problem

Let $a_0 = \dfrac{1}{2}$ and

a_k = a_{k-1} + \frac{1}{n}a_{k-1}^2 \qquad (k = 1, 2, \ldots, n).

Prove that

1 - \frac{1}{n} < a_n < 1.

Answer

Solution

Difficulty	8/10
Topics	Monotonicity, sequences, Estimation, Telescoping, inequality

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