Question

x=1/1×8+1/8×15+...+1/(7n-6)(7n+1)
n=?; {x}=15/106
[x]=?

Answer 1

 

$\displaystyle\\\text{Folosim formula: }~~~\frac{7}{n(n+7)}=\frac{1}{n}-\frac{1}{n+7}\\\\x=\frac{1}{1\times8}+\frac{1}{8\times15}+\frac{1}{15\times23}+...+\frac{1}{(7n-13)(7n-6)}+\frac{1}{(7n-6)(7n+1)}\\\\ \textbf{Inmultim egalitatea cu 7.}\\\\7x=\frac{7}{1\times8}+\frac{7}{8\times15}+\frac{7}{15\times22}+...+\frac{7}{(7n-13)(7n-6)}+\frac{7}{(7n-6)(7n+1)}$

.

$\displaystyle\\7x=\frac{1}{1}}-\frac{1}{8}+\frac{1}{8}-\frac{1}{15} +\frac{1}{15}-\frac{1}{22} +...\\\\...+\frac{1}{7n-13}-\frac{1}{7n-6}+\frac{1}{7n-6}-\frac{1}{7n+1}\\\\\text{termenii asemenea, care sunt alaturati, se reduc 2 cate 2 si ramanem cu:}\\\\7x=\frac{1}{1}}-\frac{1}{7n+1}\\\\7x=1-\frac{1}{7n+1}\\\\7x=\frac{7n+1-1}{7n+1}\\\\7x=\frac{7n}{7n+1}\\\\x=\frac{7n}{7(7n+1)}\\\\x=\frac{n}{7n+1}~~~\text{unde n este numar natural}$

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$\displaystyle\\\frac{n}{7n+1}~~\text{este subunitara}\\\\\implies~x=\{x\}\\\implies~[x]=0\\\\\{x\}=\frac{15}{106}\\\\x=\{x\}=\frac{n}{7n+1}=\frac{15}{106}\\\\\text{Din cei 2 numaratori  observam ca:}\\\\n=15\\\\\text{Verificam la numitori:}\\\\7n+1=7\times15+1=105+1=106~~~\text{Corect}\\\\Solutia:\\\\~\boxed{\bf[x]=0~~~\text{si}~~~n=15}$