Prove that sum(k) from 0 to n is n(n+1)/2, by induction

Proof by induction involves making an assumption, and using that assumption to prove that the consecutive case follows the pattern. 

The key to this is realising that most questions follow the same structure, usually involving rearranging algebra. Remember to try to see where you can use the induction step, and how you can rearrange it to make it clear how the induction step fits in. Just keep calm, write out every step carefully, and the answers will follow.

Base case: for k=1, sum(0+1) = 1 and 1(1+1)/2 = 1, and we have shown that the claim is true in this case.

Hypothesis: suppose the claim is true for k=n

Induction step: for k=n+1 , take the sum:

sum(k) [0--n+1] = sum(k)[0--n] + n+1 = n(n+1)/2 +n+1 = (n²+n)/2 + (2n+2)/2 = (n²+3n+2)/2

= (n+1)(n+2)/2 and we have shown the claim

Conclusion: As the claim is true for 0 and 1, and we have shown it to be true if it is true for n=k, by induction we have proved it true for all n in the natural numbers.