Question

2supra 1×3+2supra3×5+2supra5×7+...2supra45×47

Answer 1

$\frac{2}{1*3}+\frac{2}{3*5}+\frac{2}{5*7}+...+\frac{2}{45*47}=$

$=\frac{3-1}{1*3}+\frac{5-3}{3*5}+\frac{7-5}{5*7}+...+\frac{47-45}{45*47}=$

$=\frac{3}{1*3}-\frac{1}{1*3}+\frac{5}{3*5}-\frac{3}{3*5}+\frac{7}{5*7}-\frac{5}{5*7}+...+\frac{47}{45*47}-\frac{45}{45*47}=$

$=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{45}-\frac{1}{46}=$

$=\frac{1}{1}-\frac{1}{46}=$

$=\frac{46}{46}-\frac{1}{46}=$

$=\boxed{\frac{45}{46}}$