How Can You Derive the Formula for the Sum of the First n Integers?

[atrus_ovis]

Hello, i just came accros:

Sum(i) , from i=1 to i=n
which apparently equals n(n+1)/2

-Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation?
-Do you have any resources to offer, that includes all the common sums and its closed forms?

Thanks in advance

 

[mathman]

Very elementary proof: Let S=sum,

S = 1 + 2 + ... + n
S = n + (n-1) + ... + 1
2S = (n+1) + (n+1) + ... + (n+1) {n terms} = n(n+1)

 

[atrus_ovis]

Damn... that was way too trivial, i feel worthless =P

 

[micromass]

Hello, i just came accros:

Sum(i) , from i=1 to i=n
which apparently equals n(n+1)/2

-Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation?
-Do you have any resources to offer, that includes all the common sums and its closed forms?

Thanks in advance

Check out the very interesting paper: www.math.uic.edu/~kauffman/DCalc.pdf
This gives some tools to derive closed-form formula's of some nice sums. And these tools are based on the normal tools of calculus.

 

[Dr. Seafood]

I think these kinds of formulas are conjectured (guessed) using a clever trick or by noticing a pattern, and then formally established using mathematical induction. In the particular example of the sum of the first n integers, take n = 10:

We want the sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. Notice that 11 = 1 + 10 = 2 + 9 = 3 + 8, etc. and so we've added 11 5 = 10/2 times. The conjecture is that the sum is 11 + 11 + 11 + 11 + 11 = 5(11) = (10/2)(11). Perhaps this holds for all n: one of n or n + 1 is even since they are adjacent integers, so n(n + 1) is always divisible by two; so n(n + 1)/2 is an integer. Makes sense that the sum of the first n integers is ... We guess that 1 + 2 + ... + n = n(n + 1)/2 and establish this by induction.

This kind of guess-work is useful for a simple, intuitive arithmetic formula like this one, but probably not for more complicated expressions/conjectures.

Check out the very interesting paper: www.math.uic.edu/~kauffman/DCalc.pdf
This gives some tools to derive closed-form formula's of some nice sums. And these tools are based on the normal tools of calculus.

That is so awesome. Even as a PM major taking courses like RA, I find discrete math so beautiful -- so graph theory and combinatorics really interests me (but the combinatorics program at my school is too computer-y for me D:). It's so cool to see the methods of calculus used in a discrete setting.

 

[Mensanator]

Hello, i just came accros:

Sum(i) , from i=1 to i=n
which apparently equals n(n+1)/2

-Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation?
-Do you have any resources to offer, that includes all the common sums and its closed forms?

Thanks in advance

Goto mensanator.com, click on The Joy of Six, then click on "What Is The Sum of Integers".