You can even simply the thought behind this further by simply taking number 1 to n and listing each number in succession and then below each number list n to 1 (backwards). Then summing down the row. You'll notice that each number summed down is simply n+1. But because we've counted each column of numbers twice, we need to divide by 2 to "normalize" the results and get the correct answer. So for example take n = 4. Then write out:
1 2 3 4
4 3 2 1
And now add from the 4 columns:
1 2 3 4
4 3 2 1
5 5 5 5
Summing this is 5+5+5+5 or [5 x 4 or 5 x (n-1)] = 20. Since we've double counted each column, we need to normalize by dividing by 2, which results in 20/2 = 10.
Notice that 1 + 2 + 3 + 4 = 10.
Nice and simple explanation thanks for that :)
Yes, this is just like the method you initially mentioned, but without the geometry explanation. Nice post! Similar induction arguments can be made to obtain simple formulas for the sum of numbers squared, cubed, etc. There is a nice body of work around these methods.