cokertaylor25
Answered

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Please I need some help!!!!!!!
The sum of the series \(\sum_{i^{-1}}^{10}5\left(\frac{1}{2}\right)^i\) is \(\frac{x}{1,024}\).

Then x =_______.

Sagot :

LammettHash

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.

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