Proving an inequality by mathematical induction

Continue with:

$(1 + \frac{k}{2}) * (1 + \frac{1}{2}) =$

$1 + \frac{k}{2} + \frac{1}{2} + \frac{k}{4} >$

$1 + \frac{k}{2} + \frac{1}{2}=$

$1 + \frac{k+1}{2}$

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.

Continue expanding the product.

$$(1 + \frac{1}{2})^{k+1} =(1 + \frac{1}{2})^k \cdot (1 + \frac{1}{2}) \ge (1+\frac{k}2)(1 + \frac{1}{2}) = 1+\frac{k}2 + \frac12+\frac{k}{4}>1+\frac{k+1}{2}$$