A Slope Condition on Sequences

Problem

Say a sequence $\{a_n\}$ has property $P(t)$ if

\frac{a_m - a_n}{m - n} \ge t \qquad \text{for all } m \ne n \in \mathbb{N}^*.

1. If $a_n = 2^n$ has property $P(t)$ , find the largest such $t$ . 2. If $a_n = n^2 - \dfrac{a}{n}$ has property $P(10)$ , find the range of $a$ .

Answer

Solution

Difficulty	6/10
Topics	Extrema, Monotonicity, sequences

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