+26367 Evaluate
\( \frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7\).
\(\begin{array}{|rcll|} \hline && \frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7 \\ &=& \frac{2+1}{2}+\frac{4+1}{4}+\frac{8+1}{8}+\frac{16+1}{16}+\frac{32+1}{32}+\frac{64+1}{64}-7 \\ &=& \frac{2}{2}+ \frac{1}{2}+ \frac{4}{4} + \frac{1}{4}+ \frac{8}{8} + \frac{1}{8}+ \frac{16}{16} \frac{1}{16}+ \frac{32}{32}+ \frac{1}{32}+ \frac{64}{64} + \frac{1}{64}-7 \\ &=& 1+ \frac{1}{2}+ 1 + \frac{1}{4}+ 1 + \frac{1}{8}+ 1 + \frac{1}{16}+ 1+ \frac{1}{32}+ 1 + \frac{1}{64}-7 \\ &=& \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16}+\frac{1}{32}+ \frac{1}{64}+6-7 \\ &=& \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16}+\frac{1}{32}+ \frac{1}{64} -1 \\ &=& \frac{32}{64}+ \frac{16}{64}+ \frac{8}{64}+ \frac{4}{64}+\frac{2}{64}+ \frac{1}{64} -1 \\ &=& \frac{1+2+4+8+16+32}{64} -1 \\ &=& \frac{64-1}{64} -1 \\ &=& \frac{64}{64} - \frac{1}{64} -1 \\ &=& 1 - \frac{1}{64} -1 \\ &=& - \frac{1}{64}\\ \hline \end{array}\)
+128521 3/2+5/4+9/8+17/16+33/32+65/64-7 =
1 + 1/2 + 1 + 1/4 + 1 + 1/8 + 1 + 1/16 + 1 + 1/32 + 1 + 1/64 - 7 =
1 + 1 + 1 + 1 + 1 + 1 - 7 + [ 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64] =
6 - 7 + (1/2) [ 1 - (1/2)^6] / [ 1 - 1/2] =
-1 + (1/2) (63/64)/ (1/2) =
-1 + 63/64 =
-64/64 + 63/64 =
-1/64
