Question

The sum of the series. Can you explain how to do? Thanks!

Answer 1

Answer:

  5115

Step-by-step explanation:

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

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

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 ...

  $S_n=a_1\dfrac{1-r^n}{1-r}\\\\S_{10}=\dfrac{5}{2}\cdot\dfrac{1-\left(\dfrac{1}{2}\right)^{10}}{1-\dfrac{1}{2}}=\dfrac{5}{2}\cdot\dfrac{1023}{512}=\dfrac{5115}{1024}$

The numerator of the fraction is ...

  x = 5115

_____

Caveat

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