1. ## An irrational sequence

A sequence of real numbers $\displaystyle (u_1,u_2,u_3,...,u_n)$ has the following properties

$\displaystyle u_1= \sqrt{2}$ , $\displaystyle u_2=\pi$

$\displaystyle u_n=u_{n-1}-u_{n-2}$ for $\displaystyle n\geq{3}$

What is the value of $\displaystyle u_{2008}$?

Thanks for the help.

2. Hello, I-Think!

A sequence of real numbers $\displaystyle \{u_1,u_2,u_3, \hdots u_n\}$ has the following properties

$\displaystyle u_1\:=\:\sqrt{2}$
$\displaystyle u_2\:=\:\pi$
$\displaystyle u_n\:=\:u_{n-1}-u_{n-2}\:\text{ for } n\geq{3}$

What is the value of $\displaystyle u_{2008}$?
Crank out the first few terms . . .

. . $\displaystyle \begin{array}{ccc} u_1 &=& {\color{red}\sqrt{2}} \\ u_2 &=& {\color{blue}\pi} \\ u_3 &=& \pi - \sqrt{2} \\ u_4 &=& \text{-}\sqrt{2} \\ u_5 &=& \text{-}\pi \\ u_6 &=& \text{-}\pi + \sqrt{2} \\ u_7 &=& {\color{red}\sqrt{2}} \\ u_8 &=& {\color{blue}\pi} \\ \vdots & & \vdots \end{array}$

The sequence "loops" through a six-step cycle.

$\displaystyle \text{Since }2008 \:=\:334(6) + 4,$

. . $\displaystyle \text{then: }\:u_{2008} \;=\;u_4 \;=\;-\sqrt{2}$