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If 1/2+1/4+1/8+......1/512+1/1024
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17 Apr 2025, 03:17
17 Apr 2025, 03:17
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If \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+......\frac{1}{512}+\frac{1}{1024}=\frac{x}{y}\), then y-x=?
A. 0
B. 1
C. 128
D. 512
E. 1023
A. 0
B. 1
C. 128
D. 512
E. 1023
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Re: If 1/2+1/4+1/8+......1/512+1/1024
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22 Apr 2025, 04:15
22 Apr 2025, 04:15
Expert Reply
The given sequence $\(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\ldots+\frac{1}{2^{10}}\)$ is in Geometric Progression (G.P).
The sum of the sequence in G.P $\(=\frac{a\left(1-r^n\right)}{(1-r)}\)$ where $\(a=\frac{1}{2}, r=\frac{1}{2}\)$ and $\(n=10\)$. Therefore, the sum of the given sequence
$$
\(=\frac{\frac{1}{2}\left(1-\left(\frac{1}{2}\right)^{10}\right)}{\left(1-\frac{1}{2}\right)}=1023 / 1024=x / y \Rightarrow y-x=1\) .
$$
Thus, the correct option is $B$.
The sum of the sequence in G.P $\(=\frac{a\left(1-r^n\right)}{(1-r)}\)$ where $\(a=\frac{1}{2}, r=\frac{1}{2}\)$ and $\(n=10\)$. Therefore, the sum of the given sequence
$$
\(=\frac{\frac{1}{2}\left(1-\left(\frac{1}{2}\right)^{10}\right)}{\left(1-\frac{1}{2}\right)}=1023 / 1024=x / y \Rightarrow y-x=1\) .
$$
Thus, the correct option is $B$.
gmatclubot
Re: If 1/2+1/4+1/8+......1/512+1/1024 [#permalink]
22 Apr 2025, 04:15
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