Prove the following given $0<a<b$.

Question

If $\frac{b}{a} < \frac{a+b}{b}$, then prove the following:

1. $\frac{b}{a} < \frac{a + 2b}{a+b}$

2. $\frac{a+b}{b} < \frac{2a + 3b}{a+2b}$

I have tried to simplify these inequalities, but have gotten no where. Please help!

Answer 1

Hint: Since \begin{align} b^2<a^2+ab \end{align} then it follows \begin{align} b^2+ab<a^2+2ab. \end{align}