Supremum of a set given by $\sum\limits_{k=1}^{n}\frac{a_{k}}{a_{k}+a_{k+1}+a_{k+2}}$ (one inequality)

Question

I've found a solution for this problem: $\inf$ and $\sup$ of a set given by $\sum\limits_{k=1}^{n}\frac{a_{k}}{a_{k}+a_{k+1}+a_{k+2}}$ but I'm somehow struggling to understand one inequality, that is $$\frac{a_{k}}{a_{k}+a_{k+1}+a_{k+2}}< \frac{\sum_{i=1}^{k}a_{i}+\sum_{i=k+3}^{n}a_{i}}{\sum_{i=1}^{n}a_{i}}$$ (I created a new topic since I cannot comment overthere). Tia.

Answer 1

Hint: If $ \frac{a}{b} < 1$, then for any $ c>0$, $\frac{a}{b} < \frac{ a+c}{b+c}$.

Apply this directly. If you're stuck, show your work and what you've tried.

Answer 2

This is a cute inequality, that comes from the following observation:

$$\frac{\sum_{i=1}^{k}a_{i}+\sum_{i=k+3}^{n}a_{i}}{\sum_{i=1}^{n}a_{i}}=1-\frac{a_{k+1}+a_{k+2}}{\sum a_i}>1-\frac{a_{k+1}+a_{k+2}}{a_k+a_{k+1}+a_{k+2}}=\frac{a_k}{a_k+a_{k+1}+a_{k+2}}$$