Answer:
[tex]S_{250} = 62750[/tex]
Explanation:
Given
[tex]Sequence: 2 + 4 + 6 + ..... + 500[/tex]
Required
Evaluate the sequence
The sequence shows an arithmetic progression and will be solved using the sum of n terms of an Arithmetic Sequence as follows:
[tex]S_n = \frac{n}{2}(a + L)[/tex]
But first, we need to determine the value of n as follows:
[tex]L = a + (n - 1)d[/tex]
Where
[tex]L = Last\ Term = 500[/tex]
[tex]a = First\ Term = 2[/tex]
[tex]d = Common\ Difference = 4 - 2 = 2[/tex]
So:
[tex]L = a + (n - 1)d[/tex]
[tex]500 = 2 + (n - 1) * 2[/tex]
Open bracket
[tex]500 = 2 + 2n - 2[/tex]
Collect Like Terms
[tex]2n = 500 + 2 - 2[/tex]
[tex]2n = 500[/tex]
Divide through by 2
[tex]n = 250[/tex]
So:
[tex]S_n = \frac{n}{2}(a + L)[/tex] becomes
[tex]S_{250} = \frac{250}{2}(2 + 500)[/tex]
[tex]S_{250} = 125(2 + 500)[/tex]
[tex]S_{250} = 125(502)[/tex]
[tex]S_{250} = 125 * 502[/tex]
[tex]S_{250} = 62750[/tex]
Hence, the sum of the sequence is 62750
