1. Sep 14, 2011

flyingpig

1. The problem statement, all variables and given/known data

Prove that for all integers $$n \geq 1$$, one has

$$1 + 2 + ... + n = \frac{n(n+1)}{2}$$

(1) S(1) = 1, true

(2) Let n = k + 1

$$1 + 2 + ... + k + (k + 1) = \frac{(k+`1)(k + 2)}{2}$$

3. The attempt at a solution

Why is the last series

$$1 + 2 + ... + k + (k +1)$$ instead of $$1 + 2 +...+ (k + 1)$$?

2. Sep 14, 2011

rock.freak667

Because the (k+1)th term to each side of the equation, which happens to be 'k+1'. The right side simplifies to (k+1)(k+2)/2

3. Sep 14, 2011

Staff: Mentor

You're skipping a step here, again. There are three things you have to establish in an induction proof:
1) Base case (typically for n = 1)
2) The induction hypothesis - you assume that the statement is true for n = k
3) The induction step (I think that's what it's called) - you use the statement for n = k to show that the statement is also true for n = k + 1.
These two are exactly the same. Each one represents the sum of the integers from 1 through k + 1. The first expression explicitly shows k, and the other one doesn't, but we can infer that the second expression doesn't skip from k - 1 to k + 1 in the sum.

4. Sep 14, 2011

flyingpig

Is it bad that I don't show it?

5. Sep 14, 2011

Staff: Mentor

If your professor is a stickler, or if he/she isn't convinced that you know what it should be, it is.

In an induction proof, you should ALWAYS write down your induction hypothesis.