Question

Calculati:
1/2+1/3+1/4+...+1/10

Answer 1

 

$\displaystyle\\\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}=?\\\\\text{Numitorul comun il calculam inmultind numerele prime }\\\text{la puterea cea mai mare pe care o gasim printre numitori:}\\\\ \text{cmmmc}=2^3\times3^2\times5^1\times7^1=8\times9\times5\times7=2520\\\\\text{Amplificam fiecare fractie cu raportul dintre 2520 si numitorul ei.}$

$\displaystyle\\ \frac{1}{2}~\text{o amplificam cu }~\frac{2520}{2}=1260\implies \frac{1}{2}=\frac{1260}{2520}\\\\ \frac{1}{3}~\text{o amplificam cu }~\frac{2520}{3}=840\implies \frac{1}{3}=\frac{840}{2520}\\\\ \frac{1}{4}~\text{o amplificam cu }~\frac{2520}{4}=630\implies \frac{1}{4}=\frac{630}{2520}\\\\ \frac{1}{5}~\text{o amplificam cu }~\frac{2520}{5}=504\implies \frac{1}{5}=\frac{504}{2520}\\\\ \frac{1}{6}~\text{o amplificam cu }~\frac{2520}{6}=420\implies \frac{1}{6}=\frac{420}{2520}$

$\displaystyle\\ \frac{1}{7}~\text{o amplificam cu }~\frac{2520}{7}=360\implies \frac{1}{7}=\frac{360}{2520}\\\\ \frac{1}{8}~\text{o amplificam cu }~\frac{2520}{8}=315\implies \frac{1}{8}=\frac{315}{2520}\\\\ \frac{1}{9}~\text{o amplificam cu }~\frac{2520}{9}=280\implies \frac{1}{9}=\frac{280}{2520}\\\\ \frac{1}{10}~\text{o amplificam cu }~\frac{2520}{10}=252\implies \frac{1}{10}=\frac{252}{2520}$

$\displaystyle\\\\\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}=\\\\=\frac{1260}{2520}+\frac{840}{2520}+\frac{630}{2520}+\frac{504}{2520}+\frac{420}{2520}+\frac{360}{2520}+\frac{315}{2520}+\frac{280}{2520}+\frac{252}{2520}=\\\\=\frac{1260+840+630+504+420+360+315+280+252}{2520}=\\\\=\boxed{\bf\frac{4861}{2520}}\\\\\text{Fractia nu se simplifica deoarece 4861 este numar prim.}$

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