Define $(b_{n})$ to be: $b_{0}=12$, $b_{1}=\frac{\sqrt{3}}{2}$ adn $b_{n+1}+b_{n-1}=b_{n}\cdot\sqrt{3}$. Find $b_{0}+b_{1}+\dots+b_{2007}$. Note. Maybe this seems too easy, but I want to post all the problems...
Problem
Source: Easy .. still, to post the complete olympiad...
Tags: algebra unsolved, algebra
Rust
04.03.2007 20:38
$b_{n}=Re(12+11\sqrt 3)exp(\frac{\pi ni}{6})$. $b_{0}+b_{1}+...+b_{2007}=b_{0}+b_{1}+b_{2}+b_{3}=-\frac{345+306\sqrt{3}}{13}$.
N.T.TUAN
07.03.2007 18:08
Rust wrote: $b_{n}=Re(12+11\sqrt 3)exp(\frac{\pi ni}{6})$. $b_{0}+b_{1}+...+b_{2007}=b_{0}+b_{1}+b_{2}+b_{3}=-\frac{345+306\sqrt{3}}{13}$. Why? I think you only have $b_{i}=b_{i+12}$
Rust
07.03.2007 19:51
N.T.TUAN wrote: Rust wrote: $b_{n}=Re(12+11\sqrt 3)exp(\frac{\pi ni}{6})$. $b_{0}+b_{1}+...+b_{2007}=b_{0}+b_{1}+b_{2}+b_{3}=-\frac{345+306\sqrt{3}}{13}$. Why? I think you only have $b_{i}=b_{i+12}$ $\sum_{i=0}^{11}b_{k+i}=0$.
computer_nuke
07.03.2007 20:27
characteristic equation and solve
Rendal
07.09.2018 09:25
Rust wrote: $b_{n}=Re(12+11\sqrt 3)exp(\frac{\pi ni}{6})$. $b_{0}+b_{1}+...+b_{2007}=b_{0}+b_{1}+b_{2}+b_{3}=-\frac{345+306\sqrt{3}}{13}$. My answer: $ \frac{3-21 \sqrt{3}}{2}$