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# Help

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What is the following value when expressed as a common fraction:

$$\frac{1}{2^{1}}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots + \frac{1}{2^{8}}+\frac{1}{2^{9}}+\frac{1}{2^{10}}?$$

Jan 5, 2021

#1
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if we convert each number to make the denominator 2^10, we get 2^9+2^8...1(all the numerators).

If we add powers of 2 starting from one, it is always one less than the nest power of 2

1+2 is 1 less than 4, 1+2+4 is 1 less than 8, and so on.

So it would be (2^10-1)/2^10, or 1023/1024.

Jan 5, 2021

#1
+412
+1