Induction

1st-year undergraduate combinatorics. 

Use induction to prove that for any natural n the following holds (see title)

Given n positive numbers $x_1, x_2, x_3, ... x_n$ such thatx_1+x_2+x_3+\dots +x_n \leq \frac{1}{3}.Prove by induction that (1-x_1)(1-x_2)(1-x_3)....(1-x_n) \geq \frac{2}{3}.

Attached is a photo of the problem. Help would be much appreciated. It involves proving an identity using t-formulas, then using that result prove the following using mathematical induction. Then, sho...

prove

Prove that \sum_{i=0}^n (f_i)^2=f_nf_{n+1},where $f_n$ is the Fibonacci sequence.