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\(\dfrac{5}{8}+\dfrac{9}{8}=\dfrac{14}{8}=\dfrac{7}{4}\)
\(\dfrac{6}{7}-\dfrac{3}{7}=\dfrac{3}{7}\)
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\(B=1+2+3+...+100\)
Total B has:
\(\dfrac{100-1}{1}+1=100\) (number)
Total B is:
\(\dfrac{\left(100+1\right)\cdot100}{2}=5050\)
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\(100+\dfrac{500}{5}-40\)
\(=100+100-40\)
\(=200-40\)
\(=160\)
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Land \(B=1.2+2.3+3.4+...+199.200\)
\(\Rightarrow3B=1.2.\left(3-0\right)+2.3.\left(4-1\right)+...+199.200.\left(201-98\right)\)
\(\Rightarrow3B=1.2.3-1.2.3+2.3.4-2.3.4+...+199.200.201\)
\(\Rightarrow3B=199.200.201\)
\(\Rightarrow B=\dfrac{199.200.201}{3}\)
\(\Rightarrow B=2666600\)
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We have:
\(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}\)
\(=\dfrac{2.1}{2.1.3}+\dfrac{2.1}{2.3.5}+\dfrac{2.1}{2.5.7}+...+\dfrac{2.1}{2.99.101}\)
\(=\dfrac{1}{2}.\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\right)\)
\(=\dfrac{1}{2}.\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)
\(=\dfrac{1}{2}.\left(1-\dfrac{1}{101}\right)\)
\(=\dfrac{1}{2}.\dfrac{100}{101}\)
\(=\dfrac{50}{101}\)
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