What is the value of \(x\)
Home -> Solved problems -> What is the value of x
Solution
\[\begin{aligned}
&x+\frac{x}{2}+\frac{x}{4}+\cdots+\frac{x}{512}+\frac{x}{1024}=1+2+4+\cdots+512+1024 \\\\
&\Leftrightarrow \quad x\left(1+\frac{1}{2}+\frac{1}{2^{2}}+\cdots+\frac{1}{2^{9}}+\frac{1}{2^{10}}\right)=1+2+2^{2}+\cdots+2^{9}+2^{10} \\\\
&\Leftrightarrow \quad x \cdot\left[1 \cdot \frac{1-\left(\frac{1}{2}\right)^{11}}{1-\frac{1}{2}}\right]=1 \cdot \frac{1-2^{11}}{1-2} \\\\
&\Leftrightarrow \quad 2 x \cdot\left(1-\frac{1}{2^{11}}\right)=2^{11}-1 \\\\
&\Leftrightarrow \quad 2 x \cdot \frac{2^{11}-1}{2^{11}}=2^{11}-1 \\\\& \Leftrightarrow x=\frac{\left(2^{11}-1\right) \cdot 2^{11}}{2 \cdot\left(2^{11}-1\right)}=2^{10}=1024
\end{aligned}\]
Home -> Solved problems -> What is the value of x
Every problem you tackle makes you smarter.
↓ Scroll down for more maths problems↓
Prove that the function \(f(x)=\frac{x^{3}+2 x^{2}+3 x+4}{x}
\) has a curvilinear asymptote \(y=x^{2}+2 x+3\)
Why does the number \(98\) disappear when writing the decimal expansion of \(\frac{1}{9801}\) ?
if we draw an infinite number of circles packed in a square using the method shown below, will the sum of circles areas approach the square's area?
