\(2B=\frac{2}{1.3}+\frac{2}{3.5}+....+\frac{2}{9.11}\)
\(2B=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{11}\)
\(2B=\frac{1}{1}-\frac{1}{11}=\frac{10}{11}\)
\(B=\frac{10}{11}:2=\frac{10}{11}.\frac{1}{2}=\frac{5}{11}\)
\(B=\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{7.9}+\frac{2}{9.11}\right)\)
\(=\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{11}\right)\)
\(=\frac{1}{2}.\frac{10}{11}=\frac{5}{11}\)
B = 1/3 + 1/15 + 1/35 + 1/63 + 1/99
B = 1/2 × (2/1×3 + 2/3×5 + 2/5×7 + 2/7×9 + 2/9×11)
B = 1/2 × (1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11)
B = 1/2 × (1 - 1/11)
B = 1/2 × 10/11
B = 5/11
\(.B=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}\)
\(\Rightarrow B=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{11}\right)\)
\(\Rightarrow B=\frac{1}{2}.\left(1-\frac{1}{11}\right)\)
\(\Rightarrow B=\frac{1}{2}.\frac{10}{11}\)
\(\Rightarrow B=\frac{10}{22}\)
B=1/3 + 1/15 +1/35 +1/63 + 1/99
=1/2x (1/1x3 +1/3x5 +1/5x7 +1/7x9 +1/9x11)
=1/2x (1 -1/3 + 1/3 -1/5 +1/5 -1/7 +1/7 -1/9 + 1/9 -1/11)
=1/2x (1 -1/11)
=1/2 x 10/11
=5/11
A = 1/31 + 1/32 + ... + 1/60
A = (1/31 + 1/32 + ... + 1/40) + (1/41 + 1/42 + ... + 50) + (1/51 + 1/52 + ... + 1/60)
A > 1/40 × 10 + 1/50 × 10 + 1/60 × 10
A > 1/4 + 1/5 + 1/6
A > 1/4 + 1/6 + 1/6
A > 1/4 + 1/3
A > 7/12
A = 1/31 + 1/32 + ... + 1/60
A = (1/31 + 1/32 + ... + 1/40) + (1/41 + 1/42 + ... + 50) + (1/51 + 1/52 + ... + 1/60)
A > 1/40 × 10 + 1/50 × 10 + 1/60 × 10
A > 1/4 + 1/5 + 1/6
A > 1/4 + 1/6 + 1/6
A > 1/4 + 1/3
A > 7/12
\(B=\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{7.9}+\frac{2}{9.11}\right)\)
\(=\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{11}\right)\)
\(=\frac{1}{2}.\frac{10}{11}=\frac{5}{11}\)
