23666: HW 4
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| − | '''Problem 16.''' Prove for all natural numbers $n\geq 5$: $2^n>n^2$. | + | '''Problem 16.''' Prove for all natural numbers $n\geq 5$:\ \ $2^n>n^2$. |
'''Problem 17.''' Let $n\in\mathbb{N}$. Conjecture a formula for the expression | '''Problem 17.''' Let $n\in\mathbb{N}$. Conjecture a formula for the expression | ||
Revision as of 15:27, 12 March 2019
Problem 16. Prove for all natural numbers $n\geq 5$:\ \ $2^n>n^2$.
Problem 17. Let $n\in\mathbb{N}$. Conjecture a formula for the expression \[a_n=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n\cdot (n+1)}\] and prove your conjecture by induction.
Problem 18. Use Problem 15 and induction to show that ${\cal P}(A)$ has $2^n$ elements, when $A$ has $n$ elements.
