which one is bigger $100^n+99^n$ or $101^n$ Suppose $n \in \mathbb{N} , n>1000$ now how can we prove :which one is bigger $$100^n+99^n \text{ or  }  101^n \text{ ? }$$ 
I tried to use $\log$ but get nothing . Then I tried for binomial expansion...but  I get stuck on this . 
can someone help me ? thanks in advance.

Of course
$$1.01^n>1+\frac{n}{100}>2$$
for $n>100$, and obviously $1+0.99^n<2$.
