Proof By Induction Inequalities Questions, Answers and Solutions

  a - answer    s - solution    v - video    d - discussion

Question 1


Prove by mathematical induction:

a) $2^n>2n$ for all $n>2$ and $n\in\mathbb{Z}^+$a s v d

b) $2^n<3^n$ for $n\geq1$ and $n\in\mathbb{Z}^+$a s v d

c) $n!>2^n$ for $n\geq4$a s v d

d) $2^n>4n$ for $n\geq5$ and $n\in\mathbb{Z}^+$a s v d

e) $n^2\ge2n$ for $n>1$ and $n\in\mathbb{Z}^+$a s v d

f) $n^2<4^{n-1}$ for $n\geq3$ and $n\in\mathbb{Z}^+$a s v d



Question 2


Prove by mathematical induction:

a) $1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}\dotsb+\frac{1}{\sqrt n}<2\sqrt n-1$ for $n\ge2$ and $n\in\mathbb{Z}^+$a s v d

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