Why write 1+2+...+k+(k+1) instead of 1+2+...+(k+1) in induction

[flyingpig]

Homework Statement

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

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

The Attempt at a Solution

Why is the last series

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

 

[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

 

[Mark44]

Homework Statement

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

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.

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

The Attempt at a Solution

Why is the last series

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

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.

 

[flyingpig]

Is it bad that I don't show it?

 

[Mark44]

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.