Prove that ${n\choose0} + {n\choose3} + {n\choose6} + …. + {n\choose3k} \le \frac {1}{3} (2^n + 2)$ where $n$ is a positive integer and $k$ is the largest integer for which $3k \le n.$
I tried induction and came till where I have to prove ${n\choose2} + {n\choose5} + ……+ {n\choose3k-1} \le {n\choose0} + {n\choose3} + ……+ {n\choose3k}.$
How to do it? And is there any better method of proving this question?
