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 | ||
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'''Problem 18.''' Use Problem 15 and induction to show that ${\cal P}(A)$ has $2^n$ elements, when $A$ has $n$ elements. | '''Problem 18.''' Use Problem 15 and induction to show that ${\cal P}(A)$ has $2^n$ elements, when $A$ has $n$ elements. | ||
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| + | '''Problem 19.''' Show that every natural number greater than 33 can be written in the form $4s+5t$, where $s$ and $t$ are natural numbers with $s\geq3$ and $t\geq 2$. | ||
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| + | '''Problem 20.''' Let $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$ for all $n\geq 1$. Show that $a_n=2^n$ for all natural numbers $n$. | ||
Latest revision as of 15:35, 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.
Problem 19. Show that every natural number greater than 33 can be written in the form $4s+5t$, where $s$ and $t$ are natural numbers with $s\geq3$ and $t\geq 2$.
Problem 20. Let $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$ for all $n\geq 1$. Show that $a_n=2^n$ for all natural numbers $n$.
