Prove by induction the sum of the natural numbers from 1 to n is n(n+1)/2

Need whiteboard throughout to properly answer, so will go through the ideas of what to do:

Take the base case of when n=1, and show that            (sum from 1 to 1) j = n(n+1)/2       is equal to 1.

Take the assumption that this is true for some n=k in the natural numbers. So want to show it's true for n=k+1.

Use the sigma notation to split the sum from 1 to k+1 to the sum from 1 to k, and adding k+1. We have assumed that

(sum from 1 to k) j = k(k+1)/2, so we now have that (sum from 1 to k) j + (k+1) = k(k+1)/2 + (k+1), and we can then show that this is equal to (k+1)(k+2)/2.

So since we have shown that the statement is true for a base case, and that if it is true for n=k, it is also true for n=k+1, then we have proved the statement by the mathematical principle of induction.