Sum portion of array without loop

Hi,

I do hope the attached picture will be sufficient in order to help me.
(I faced a problem to attach this small WB as an XLS).

I'm looking for some way to some the red cells without looping.

In this example the array was filled with A1:A10 values.
In the real situation the array gets its values from other source than a Worksheet Range.

As you can see I manged to transport the Array Values to F1:F10 and from here I could calculate the sum of F3:F8 but I do not want to use any helper columns.

As you can see the wrong! sum command, I have used ,presents an 'Error 9'.

Thanks, Elm