Question

Cho \(\frac{1+2+3+...+a}{a}< \frac{1+2+3+...+b}{b}\). Hãy so sánh a và b

Answer 1

Ta có công thức: \(1+2+3+4+...+n=\frac{n\cdot\left(n+1\right)}{2}\)

Ta có:\(\frac{1+2+3+...+a}{a}< \frac{1+2+3+...+b}{b}\)

\(\Leftrightarrow\frac{\frac{a\left(a+1\right)}{2}}{a}< \frac{\frac{b\left(b+1\right)}{2}}{b}\)

\(\Leftrightarrow\frac{a\left(a+1\right)}{2a}< \frac{b\left(b+1\right)}{2b}\)

\(\Leftrightarrow\frac{a+1}{2}< \frac{b+1}{2}\)

\(\Leftrightarrow a+1< b+1\)

\(\Leftrightarrow a< b\)