It looks like you're given
[tex]\displaystyle \sum_{i=1}^{10} 5 \left(\frac12\right)^i = \frac{x}{1024}[/tex]
Consider the geometric sum,
[tex]\displaystyle S = \sum_{i=1}^{10} \left(\frac12\right)^i[/tex]
[tex]S = \dfrac12 + \dfrac1{2^2} + \dfrac1{2^3} + \cdots + \dfrac1{2^{10}}[/tex]
Multiply both sides by 1/2 :
[tex]\dfrac12 S = \dfrac1{2^2} + \dfrac1{2^3} + \dfrac1{2^4} + \cdots + \dfrac1{2^{11}}[/tex]
Subtract this from S :
[tex]S - \dfrac12 S = \dfrac12 - \dfrac1{2^{11}}[/tex]
Solve for S :
[tex]\dfrac12 S = \dfrac12 - \dfrac1{2^{11}}[/tex]
[tex]S = 1 - \dfrac1{2^{10}}[/tex]
[tex]S = \dfrac{2^{10} - 1}{2^{10}}[/tex]
[tex]S = \dfrac{1023}{1024}[/tex]
The given sum is just 5 times S, so x = 1023 × 5 = 5115.
