<s>[INST] Proving an inequality by mathematical induction 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.
 [/INST] Continue expanding the product.
$$(1 + \frac{1}{2})^{k+1} =(1 + \frac{1}{2})^k \cdot (1 + \frac{1}{2}) \ge (1+\frac{k}2)(1 + \frac{1}{2}) = 1+\frac{k}2 + \frac12+\frac{k}{4}>1+\frac{k+1}{2}$$
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