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A = \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{8}\)+ \(\dfrac{1}{16}\) + \(\dfrac{1}{32}\) + \(\dfrac{1}{64}\)+......+\(\dfrac{1}{1024}\)
A \(\times\)2 = 1 + \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{8}\) + \(\dfrac{1}{16}\) + \(\dfrac{1}{32}\)+\(\dfrac{1}{64}\)+...+\(\dfrac{1}{512}\)
A\(\times\)2 - A = 1 - \(\dfrac{1}{1024}\)
A = \(\dfrac{1023}{1024}\)
B = \(\dfrac{1}{1\times2}\)+\(\dfrac{1}{2\times3}\)+\(\dfrac{1}{3\times4}\)+\(\dfrac{1}{4\times5}\)+...+\(\dfrac{1}{98\times99}\)+\(\dfrac{1}{99\times100}\)
B = \(\dfrac{1}{1}\) - \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) - \(\dfrac{1}{3}\)+ \(\dfrac{1}{3}\)-\(\dfrac{1}{4}\)+\(\dfrac{1}{4}\)-\(\dfrac{1}{5}\)+...+\(\dfrac{1}{98}\)-\(\dfrac{1}{99}\)+\(\dfrac{1}{99}\)-\(\dfrac{1}{100}\)
B = 1 - \(\dfrac{1}{100}\)
B = \(\dfrac{99}{100}\)
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