A = 1/3×5 + 1/5×7 + 1/7×9 + 1/9×11 + ... + 1/29×31
2A = 2/3×5 + 2/5×7 + 2/7×9 + 2/9×11 + ... + 2/29×31
2A = 1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11 + .... + 1/29 - 1/31
2A = 1/3 - 1/31
2A = 31/93 - 3/93 = 28/93
A = 28/93 : 2
A = 28/93 × 1/2 = 14/93
B = 2/1×4 + 2/4×7 + 2/7×10 + ... + 2/31×34
3/2B = 3/1×4 + 3/4×7 + 3/7×10 + ... + 3/31×34
3/2B = 1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + ... + 1/31 - 1/34
3/2B = 1 -1/34 = 33/34
B = 33/34 : 3/2
B = 33/34 × 2/3 = 11/17
\(A=\frac{1}{3\times5}+\frac{1}{5\times7}+\frac{1}{7\times9}+...+\frac{1}{29\times31}\)
\(=\frac{1}{2}\times\left(\frac{1}{3}-\frac{1}{5}\right)+\frac{1}{2}\times\left(\frac{1}{5}-\frac{1}{7}\right)+...+\frac{1}{2}\times\left(\frac{1}{29}-\frac{1}{31}\right)\)
\(=\frac{1}{2}\times\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{29}-\frac{1}{31}\right)\)
\(=\frac{1}{2}\times\left(\frac{1}{3}-\frac{1}{31}\right)\)
\(=\frac{1}{2}\times\frac{28}{93}\)
\(=\frac{14}{93}\)
\(B=\frac{2}{1\times4}+\frac{2}{4\times7}+...+\frac{2}{31\times34}\)
\(=\frac{1}{3}\times\left(\frac{2}{1}-\frac{2}{4}\right)+\frac{1}{3}\times\left(\frac{2}{4}-\frac{2}{7}\right)+...+\frac{1}{3}\times\left(\frac{2}{31}-\frac{2}{34}\right)\)
\(=\frac{1}{3}\times\left(2-\frac{2}{4}+\frac{2}{4}-\frac{2}{7}+...+\frac{2}{31}-\frac{2}{34}\right)\)
\(=\frac{1}{3}\times\left(2-\frac{2}{34}\right)\)
\(=\frac{1}{3}\times\frac{33}{17}\)
\(=\frac{11}{17}\)
