Question

кто шарит, помогите пожалуйста)

Answer 1

Ответ:

Объяснение:

$\frac{1}{2*7}+\frac{1}{7*12} + \frac{1}{12*17} +\frac{1}{17*22} +...+ \frac{1}{(5n-3)(5n+2)} =$

$=\frac{1}{5} *(\frac{1}{2}-\frac{1}{7} ) + \frac{1}{5} *(\frac{1}{7}-\frac{1}{12} )+\frac{1}{5} *(\frac{1}{12}-\frac{1}{17} ) +\frac{1}{5} *(\frac{1}{17}-\frac{1}{22} ) +\frac{1}{5} *(\frac{1}{5n-3}-\frac{1}{5n+2} ) =$

$=\frac{1}{5} *(\frac{1}{2}-\frac{1}{7}+\frac{1}{7} -\frac{1}{9} +\frac{1}{9} -\frac{1}{12} +...+\frac{1}{5n-3}-\frac{1}{5n+2} ) = \frac{1}{5} *(\frac{1}{2} -\frac{1}{5n+2} ) =\frac{1}{10} -\frac{1}{25n+10}$

2.

$\frac{3}{1*2} +\frac{13}{2*3}+\frac{37}{3*4} +\frac{81}{4*5} +\frac{151}{5*6} +...+\frac{n^3+n^2+1}{n(n+1)} =$

= $\frac{3}{2} (1-\frac{1}{2}) +\frac{13}{2} (\frac{1}{2}-\frac{1}{3}) +\frac{37}{2}( \frac{1}{3 } -\frac{1}{4} )+ \frac{81}{2}(\frac{1}{4}-\frac{1}{5} ) +\frac{151}{2} (\frac{1}{5}-\frac{1}{6} )+...+\frac{n^3+n^2+1}{2} (\frac{1}{n} -\frac{1}{n+1})=$

$=\frac{3}{2} +\frac{10}{4}+\frac{24}{6} +\frac{44}{8} +\frac{70}{10} +...+1.5(n-2)+2.5 -\frac{n^3+n^2+1}{n+1}= \frac{3}{2} + (1+\frac{3}{4}n)(n-1) - \frac{n^3+n^2+1}{n+1}$