\(\dfrac{3}{2}+\dfrac{3}{8}+\dfrac{3}{32}+\dfrac{3}{128}+\dfrac{3}{512}\)
\(\dfrac{15}{8}+\dfrac{3}{32}+\dfrac{3}{128}+\dfrac{3}{512}\)
\(\dfrac{63}{32}+\dfrac{3}{128}+\dfrac{3}{512}\)
\(\dfrac{255}{128}+\dfrac{3}{512}\)
\(\dfrac{1023}{512}\)
\(=\dfrac{3}{2}+\dfrac{3}{2.4}+\dfrac{3}{4.8}+\dfrac{3}{8.16}+\dfrac{3}{16.32}\)
\(=\dfrac{3}{2}+\dfrac{3}{2}-\dfrac{3}{4}+\dfrac{3}{4}-\dfrac{3}{8}+\dfrac{3}{8}-\dfrac{3}{16}+\dfrac{3}{16}-\dfrac{3}{32}\)
\(=\dfrac{3}{2}-\dfrac{3}{32}=\dfrac{45}{32}\)
d) \(\dfrac{3}{2}+\dfrac{3}{8}+\dfrac{3}{32}+\dfrac{3}{128}+\dfrac{3}{512}\)
\(=\dfrac{3}{1\times2}+\dfrac{3}{2\times4}+\dfrac{3}{4\times8}+\dfrac{3}{8\times16}+\dfrac{3}{16\times32}\)
\(=3\left(\dfrac{1}{1\times2}+\dfrac{1}{2\times4}+\dfrac{1}{4\times8}+\dfrac{1}{8\times16}+\dfrac{1}{16\times32}\right)\)
\(=3\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{16}+\dfrac{1}{16}-\dfrac{1}{32}\right)\)
\(=3\left(1-\dfrac{1}{32}\right)\)
\(=3\times\dfrac{31}{32}\)
\(=\dfrac{93}{32}\)
