Problem: 32+54+98+1716+3332+6564−7=\dfrac{3}{2}+\dfrac{5}{4}+\dfrac{9}{8}+\dfrac{17}{16}+\dfrac{33}{32}+\dfrac{65}{64}-7=23+45+89+1617+3233+6465−7=
Answer Choices:
A. −164-\dfrac{1}{64}−641
B. −116-\dfrac{1}{16}−161
C. 000
D. 116\dfrac{1}{16}161
E. 164\dfrac{1}{64}641 Solution:
Add and subtract directly or note that
32+54+⋯+6564−7=(1+12)+(1+14)+⋯+(1+164)−7=12+14+⋯+164−1=−164\begin{aligned} \dfrac{3}{2}+\dfrac{5}{4}+\cdots+\dfrac{65}{64}-7 & =\left(1+\dfrac{1}{2}\right)+\left(1+\dfrac{1}{4}\right)+\cdots+\left(1+\dfrac{1}{64}\right)-7 \\ & =\dfrac{1}{2}+\dfrac{1}{4}+\cdots+\dfrac{1}{64}-1=-\dfrac{1}{64} \end{aligned} 23+45+⋯+6465−7=(1+21)+(1+41)+⋯+(1+641)−7=21+41+⋯+641−1=−641