TITLE: Proving an inequality by mathematical induction
QUESTION [9 upvotes]: I'm trying to solve a problem with inequalities using mathematical induction but I am stuck halfway through the process.
The problem: Use mathematical induction to establish the inequality -
$(1 + \frac{1}{2})^n \ge 1 + \frac{n}{2}$ for n $\in \mathbb{N}$
Steps
1) $n = 1$, $(1 + \frac{1}{2})^1 \ge 1 + \frac{1}{2}$ is TRUE
2) $n = k$, assume that $(1 + \frac{1}{2})^k \ge 1 + \frac{k}{2}$ for n $\in \mathbb{N}$
3) Show the statement is true for $k + 1$
$(1 + \frac{1}{2})^{k+1}$ = $(1 + \frac{1}{2})^k * (1 + \frac{1}{2})$
$\ge$ $(1 + \frac{k}{2}) * (1 + \frac{1}{2})$  - using the assumption in step $2$
My question is, how do I continue this problem? Or did I go wrong somewhere? I just can't figure out what the next step is.

REPLY [4 votes]: You are trying to establish, from $(1+1/2)^k \ge 1 + k/2$ that $(1 + 1/2)^{k+1} \ge 1 + (k+1)/2$.  That is, you are given a statement of the form:
$$a \ge b$$
and are trying to establish a statement of the form $$a\cdot c \ge d$$
So you need to establish $b \ge \frac dc$, that is, you need to establish:
$$1 + \frac k2 \ge \dfrac{1 + \dfrac{k + 1}{2}}{1 + \dfrac 12}$$
Should be straightforward.