# Mathematical proof

1. Jun 29, 2008

### ritwik06

1. The problem statement, all variables and given/known data
The thing is that there is a question which needs me to prove something. I have done it already but the thing that troubles me is that it wants me to prove this:

$$a_{1}+a_{2}+a_{3}+.................a_{K}>K$$
$$a_{i}$$ is a positive integer.
I know this is always true. but how should I prove it mathematically?

Last edited: Jun 29, 2008
2. Jun 29, 2008

### Hurkyl

Staff Emeritus
I see two ways to proceed:

(1) Attempt to precisely elaborate why you know it's true. Then translate those precise reasons into logical implications

(2) Try to prove it for special cases of your own choosing. Then, see if you can generalize your proof to the general case.

3. Jul 2, 2008

### GreenBeret

that's not true always .

example :

take $$a_i=1$$ , for all $$i$$.

So , we get $$\overbrace{1+1+\cdots+1}^K =K \not > K$$

4. Jul 3, 2008

### HallsofIvy

Staff Emeritus
I presume you mean $a_1+ a_2+ \cdot\cdot\cdot+ a_K\ge K$ where the an are positive integers.

Looks like a good candidate for "proof by induction".

5. Jul 3, 2008

### foxjwill

why proof by induction? The way I was thinking of it is like this:
Rewriting each $$a_i$$ as $$1+c_i$$, we have
$$(1+c_1)+(1+c_2)+\cdots+(1+c_K)=K+c_1+c_2+\cdots+c_K=K+C \geq K$$​

6. Jul 4, 2008

### kenewbie

This seems self-evident? The lowest positive integer is 1. If you have n numbers, all being positive, the sum cannot be smaller than n.

k