What is the sum of all even numbers from 0 to 2n, for any positive integer n?

Posted on 2011-04-24
Last Modified: 2012-06-22
The instructor from my online course went over this but unfortunately I couldn't understand his answer due to his think accent.

What is the sum of all even numbers from 0 to 2n, for any positive integer n?
Question by:Eindoofus
    LVL 31

    Expert Comment

    sum(k) for k = 1..L =    L (L+1) / 2

    But you want

    sum (2k) for k = 1..2N
    = 2 + 4+ 6 + ... + 2N
    = 2( 1 + 2 + 3 + ... + N)

    So, the rest should be easy to finish.

    Author Comment

    I believe he said that it was n(n+1) ..Is that correct?
    LVL 37

    Accepted Solution

    Yes. That is correct. Do you know what the sum of 1 + 2 + 3 + 4 + 5 + ... + N is?
    Here's how he most likely explained it.

    Let's say S = 1+2+3+4+5+...+N
    So 2S = 1+2+3+4+5+...+N  +  1+2+3+4+5+...+N
    Now flip one of them
    So 2S = 1+   2+      3     + 4+5+...+N
              +  N+(N-1)+(N-2).             .. +1
    Notice how the first one in each is N and 1, the next two are (N-1) and 2 which add to N+1
    If fact, each pair equals N+1 and there are N of them.
    So 2S = N(N+1) and S = N(N+1)/2
    This is a common result that you will be expected to remember probably.

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