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

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

hihihi Jan 5, 2021

#1**+1 **

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.

MooMooooMooM Jan 5, 2021

#1**+1 **

Best Answer

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.

MooMooooMooM Jan 5, 2021