Question

Here is another one (a lot easier):

Answer 1

http://midcitiesmathcircle.files.wordpress.com/2012/09/math_circle_fibonacci.pdf

Answer 2

try doing 1 with induction

Answer 3

try without induction

Answer 4

is that a command?

Answer 5

or a statement of what is to come?

Answer 6

with induction it seems it can be done in 3 steps. though i never tried to prove that formula.

Answer 7

I see ,,, without induction, this is just sum of geometric series.

Answer 8

Hold on, have to eat lunch, Ill see what you guys come up with when I get back

Answer 9

bunch of interesting Questions!

Answer 10

I can give you the website

Answer 11

http://midcitiesmathcircle.org/schedule/

Answer 12

nice!!

Answer 13

problem 2: just a try
\[ a_n = {\alpha^n - \beta^n \over \sqrt 5 }
\\
{1 \over 2^n \sqrt 5 } {(1 + \sqrt 5)^n - (1 - \sqrt 5)^n}\\
\]