Prove $\frac{a}c = \frac{a-b}{b-c}$ Suppose  $\frac{1}a,\frac{1}b,\frac{1}c$ are three consecutive terms in an arithmetic sequence. Show that:
$$\frac{a}c = \frac{a-b}{b-c} $$
and that:
$$\frac{2ac}{a+c} = b$$
How would I prove this?

$\dfrac{a}{c}=\dfrac{a-b}{b-c}\impliedby ab-ac=ac-bc\impliedby \color{red}{ab+bc=2ac}\impliedby\dfrac{1}{c}+\dfrac{1}{a}=\dfrac{2}{b}$ 
The second part follows from the colored part.
