You can see them on the last pages of the dictionary.

\(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\dfrac{1}{3}\)

\(=\left(\dfrac{1}{2}+\dfrac{3}{4}\right)+\left(\dfrac{2}{3}+\dfrac{1}{3}\right)\)

\(=\left(\dfrac{2}{4}+\dfrac{3}{4}\right)+1\)

\(=\dfrac{5}{4}+1\)

\(=\dfrac{9}{4}\)

Dao Trong Luan's answer is faster than mine, he's doing just right. ;)

"First month", not "month 1". We can also call it is "January". There are 31 days in January.

\(S=1+3+3^2+...+3^{50}\)

\(\Rightarrow3S=3+3^2+3^3+...+3^{51}\)

\(\Rightarrow3S-S=2S=3^{51}-1\)

So \(S=\dfrac{3^{51}-1}{2}\)

\(A=\dfrac{3\left|x\right|+2}{4\left|x\right|-5}\)

\(\Leftrightarrow4A=\dfrac{12\left|x\right|+8}{4\left|x\right|-5}\)

\(\Leftrightarrow4A=\dfrac{3\left(4\left|x\right|-5\right)+23}{4\left|x\right|-5}\)

\(\Leftrightarrow4A=\dfrac{3\left(4\left|x\right|-5\right)}{4\left|x\right|-5}+\dfrac{23}{4\left|x\right|-5}\)

\(\Leftrightarrow4A=3+\dfrac{23}{4\left|x\right|-5}\)

We have \(\left|x\right|\ge0\Rightarrow4\left|x\right|\ge0\Rightarrow4\left|x\right|-5\ge-5\)

\(\Rightarrow\dfrac{1}{4\left|x\right|-5}\le-\dfrac{1}{5}\)

\(\Rightarrow\dfrac{23}{4\left|x\right|-5}\le-\dfrac{23}{5}\)

\(\Rightarrow3+\dfrac{23}{4\left|x\right|-5}\le3+\left(-\dfrac{23}{5}\right)\)

\(\Rightarrow4A\le-\dfrac{8}{5}\)

\(\Rightarrow A\le-\dfrac{2}{5}\)

\(\Rightarrow MaxA=-\dfrac{2}{5}\) when \(x=0\)