Show that for every positive integer n, 1+2+3+⋯+n=2n(n+1).
There is a proof by induction, a proof that pairs the terms from the two ends, and a proof that stacks the numbers into a staircase of dots and counts them twice.
All three are correct. Which one tells you why there is an n+1 in the formula?
Let’s say n is 100, for this example. Write out the sum of every term from 1 to n:
1,2,3,…98,99,100
Now, take the sum of the 1st and nth term.
1+100=101
The sum is n + 1. Now, take the sum of the 2nd and nth - 1 term.
2+99=1+100+1−1=101
This relation holds true for all values in the sum and holds true for all values of n, and, since there are n values, there will be n/2 pairs. Since there are n/2 pairs of (n+1), the formula may be written