\(\frac{1}{2}+\frac{5}{6}+...+\frac{89}{90}+\frac{109}{110}\)
\(=1-\frac{1}{2}+1-\frac{1}{6}+...+1-\frac{1}{90}+1-\frac{1}{110}\)
\(=\left(1+1+...+1\right)-\left(\frac{1}{2}+\frac{1}{6}+...+\frac{1}{90}+\frac{1}{110}\right)\)(có 10 số 1)
\(=10+\left(\frac{1}{1x2}+\frac{1}{2x3}+...+\frac{1}{9x10}+\frac{1}{10x11}\right)\)
\(=10+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{10}-\frac{1}{11}\right)\)
\(=10+\left(1-\frac{1}{11}\right)\)
\(=10+\frac{10}{11}=10\frac{10}{11}\)
1/2+5/6+...+89/90+109
=(1-1/2)+(1-1/6)+...+(1-1/90)+109
=(1+1+....+1)- (1/2+1/6+...+1/90)+109 (có 9 số hạng 1)
= 9- (1/1*2+1/2*3+...+1/9*10)+109
= 9- (1-1/2+1/2-1/3+...+1/9-1/10)+109
= 9- (1-1/10) +109
=9-9/10+109
=90/10-9/10+109
=81/10+109
=1171/10