Fun math problems

gistfile1.md

Fun problems (rendered at http://mathb.in/18258)

1. Find $\displaystyle\sum_{n=1}^\infty \frac{n}{2^n}$

2. Find $\displaystyle\sum_{n=1}^\infty \frac{n^2}{2^n}$

3. Let $a(k) = \displaystyle\sum_{n=1}^\infty \frac{n^k}{2^n}$. Find a recurrence relation for $a(k)$.

4. Let $\mathcal{N}$ be the set of natural numbers that do not contain a 6 in their decimal expansion (so, $\mathcal{N} = { 1, 2, 3, 4, 5, 7, \ldots, 14, 15, 17, \ldots }$). Prove that $$ \sum_{n\in\mathcal{N}} \frac{1}{n} < 80 $$

5. Take $\mathbb{R}^2$ with the usual metric $d((x_1,y_1), (x_2,y_2)) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 }$ and randomly assign each point $(x,y)$ to a set $R$ or a set $B$ (equivalently, give each point a unique color, red or blue). Let $D_R = { d(v,w) : v,w \in R }$ and similarly let $D_B = { d(v,w) : v,w \in B }$. Prove that one or both of $D_R, D_B$ contain every non-negative real number.

6. Find unique numbers for each of the letters in the following equation so that the resulting sum holds: $$ \begin{matrix}
   & S & E & N & D \ + & M & O & R & E \ \hline M & O & N & E & Y \ \end{matrix} $$ (Note: $M \neq 0$)