23666: HW 4

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 $n$.