Question

Solve for $2\left(1-\frac{1}{2}\right)+3\left(1-\frac{1}{3}\right)+4\left(1-\frac{1}{4}\right)+\ldots

Answer 1

Answer:

$2\left(1-\frac{1}{2}\right)+3\left(1-\frac{1}{3}\right)+4\left(1-\frac{1}{4}\right)+\ldots+10\left(1-\frac{1}{10}\right)=?$

$2(\frac{2}{2} -\frac{1}{2} )+3(\frac{3}{3} -\frac{1}{3} )+4(\frac{4}{4} -\frac{1}{4} )+5(\frac{5}{5} -\frac{1}{5} )+6(\frac{6}{6}-\frac{1}{6} )+7(\frac{7}{7}-\frac{1}{7} )+8(\frac{8}{8} -\frac{1}{8} )+9(\frac{9}{9} -\frac{1}{9} )+10(\frac{10}{10} -\frac{1}{10} )\\\\=2(\frac{1}{2}) +3(\frac{2}{3})+4(\frac{3}{4})+5(\frac{4}{5}) +6(\frac{5}{6}) +7(\frac{6}{7}) +8(\frac{7}{8}) +9(\frac{8}{9}) +10(\frac{9}{10})\\\\=1+2+3+4+5+6+7+8+9\\\\=45$