Exercises: Induction: inequalities
Practice★☆☆
Prove by induction that $2^n > n$ for all $n\in\mathbb{N}$.
Prove by induction that $2^n > n$ for all $n\in\mathbb{N}$.
Practice★☆☆
Prove by induction that $3^n \geq 2n + 1$ for all $n \in \mathbb{N}$.
Prove by induction that $3^n \geq 2n + 1$ for all $n \in \mathbb{N}$.
Practice★☆☆
Prove by induction that $n! \geq 2^{n-1}$ for all $n \in \mathbb{N}$.
Prove by induction that $n! \geq 2^{n-1}$ for all $n \in \mathbb{N}$.
Practice★★☆
Prove by induction that $2^n > n^2$ for all integers $n \geq 5$.
Prove by induction that $2^n > n^2$ for all integers $n \geq 5$.
Practice★★☆
Prove by induction that $(1+x)^n \geq 1 + nx$ for all $n \in \mathbb{N}$, where $x > -1$ is a fixed real number.
Prove by induction that $(1+x)^n \geq 1 + nx$ for all $n \in \mathbb{N}$, where $x > -1$ is a fixed real number.
Practice★★☆
Prove by induction that $n! > 3^n$ for all integers $n \geq 7$.
Prove by induction that $n! > 3^n$ for all integers $n \geq 7$.
Practice★★★
Prove by induction that $$\sum_{r=1}^{n}\frac{1}{r^2} \leq 2 - \frac1n$$ for all $n\in\mathbb{N}$.
Prove by induction that
$$\sum_{r=1}^{n}\frac{1}{r^2} \leq 2 - \frac1n$$for all $n\in\mathbb{N}$.
Practice★★★
Prove by induction that $\dfrac{1}{\sqrt1} + \dfrac{1}{\sqrt2} + \dots + \dfrac{1}{\sqrt n} \geq \sqrt n$ for all $n \in \mathbb{N}$.
Prove by induction that $\dfrac{1}{\sqrt1} + \dfrac{1}{\sqrt2} + \dots + \dfrac{1}{\sqrt n} \geq \sqrt n$ for all $n \in \mathbb{N}$.
Practice★★★
a) Find the smallest positive integer $N$ for which $3^n > n^3$ for all $n \geq N$ appears to hold, by checking small values. b) Prove by induction that $3^n > n^3$ for…
- a) Find the smallest positive integer $N$ for which $3^n > n^3$ for all $n \geq N$ appears to hold, by checking small values.
- b) Prove by induction that $3^n > n^3$ for all integers $n \geq N$.
- c) Explain why the statement is false for $n = 3$.