How to show $(1+a)^n<1+2^n a$ for all $n\in\mathbb N$ and $a\in (0, 1)$?

Question

There is an induction problem which is baffling me. I'm supposed to use induction to show the inequality $$(1+a)^n< 1+2^n a,$$ for all $n\in\mathbb N$ and $a\in (0, 1)$. I guess there must be some trick to get this inequality for if I just follow the standard induction steps it does not work.

Thanks

Answer 1

Let's assume the inequality holds for all $n \in \{1,2,\ldots, k\}$. We know $$(1+a)^k<1+a2^k \tag 1$$ and it should be clear that $(1+a)^k<2^k$ as $(1+a)<2$, so multiplying through by $a$ yields $$a(1+a)^k<a2^k \tag 2$$ The sum of $(1)$ and $(2)$ gets us $$(1+a)^k+a(1+a)^k<1+a2^k+a2^k $$ and with a little algebra you will get the desired result.