Question

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 =_______.

Answer 1

It looks like you're given

`\displaystyle \sum_(i=1)^(10) 5 \left(\frac12\right)^i = (x)/(1024)`

Consider the geometric sum,

`\displaystyle S = \sum_(i=1)^(10) \left(\frac12\right)^i`

`S = \frac12 + \frac1{2^2} + \frac1{2^3} + \cdots + \frac1{2^(10)}`

Multiply both sides by 1/2 :

`\frac12 S = \frac1{2^2} + \frac1{2^3} + \frac1{2^4} + \cdots + \frac1{2^(11)}`

Subtract this from S :

`S - \frac12 S = \frac12 - \frac1{2^(11)}`

Solve for S :

`\frac12 S = \frac12 - \frac1{2^(11)}`

`S = 1 - \frac1{2^(10)}`

`S = (2^(10) - 1)/(2^(10))`

`S = (1023)/(1024)`

The given sum is just 5 times S, so x = 1023 × 5 = 5115.