23666: HW 4

Latest revision as of 16:35, 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

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 $n$.