Consider $(1+n)^k$. Where both n, k are natural numbers.
We have binomial expansion $\sum_{i=0}^{k} \binom{k}{i}n^i $
Then we have for each i-th term, certain powers of n(at least $>=i$).
As i increases, will powers of n increase too(not necessarily strictly)?
This seems trivial, but I don't see how to ensure it.
p. s. Or under what specific conditions on k will this hypothesis be true?
Thanks in advance.
It is not true:
$$\binom{6}{3}2^3 = 5*2^5\qquad \binom{6}{4}2^4 = 15*2^4$$
A sufficient condition for it to happen is that $k<n^2$.
Notice that if $k+1$ and $n$ are coprime and $k\ge n^2$, then there exists a counterexample.