Prove that 1 + 2 + · · · + n = n(n + 1)/2 for all integers n ≥ 1.

Question: Prove that 1 + 2 + · · · + n = n(n + 1)/2 for all integers n ≥ 1.

Proof:

We shall prove by induction. Base case: when n = 1 Since \[1 = 1(1 + 1)/2 \] it is true. Inductive step: Suppose the statement is true for n = k. \[1 + 2 + · · · + k = k(k+1)/2\] We shall prove that the statement is true for n = k+1 \[1+2+· · ·+k+(k+1) = (k + 1)(k + 2)/2\]. By the induction hypothesis the statement is true for n = k therefore \[1 + 2 + · · · + k + (k + 1) = k(k + 1)/2 + (k + 1).\] \[ = (k + 1)(k/2 + 1)\] \[ = (k + 1)(k + 2)/2.\] This shows that statement is true for n=k. Thus the statement is true for all integers n ≥ 1 i.e \[1 + 2 + · · · + n = n(n + 1)/2\] for all integers n ≥ 1