Question

The sum of the series.

Can you explain how to do? Thanks!

Answer 1

5115

Explanation:

First of all, we need to assume there is a typo involved, and that the expression is supposed to be ...

`\displaystyle\sum_(i=1)^(10){5\left((1)/(2)\right)^i}`

The values being summed look a lot like the general term of a geometric sequence:

`a_1(r^(n-1))`

The exponents in the sum range from 1 to 10, so the values being summed range from 5/2 to 5/1024. In order for that to be the case with our general term, for r = 1/2, we must have a1 = 5/2 as we let n range from 1 to 10.

The sum of the terms of the geometric sequence is ...

`S_n=a_1(1-r^n)/(1-r)\\\\S_(10)=(5)/(2)\cdot(1-\left((1)/(2)\right)^(10))/(1-(1)/(2))=(5)/(2)\cdot(1023)/(512)=(5115)/(1024)`

The numerator of the fraction is ...

x = 5115

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Caveat

If we have misinterpreted the intent of the problem statement, the answer will be different.