Math Proof Homework: Proving a_1 + a_2 + ... + a_K > K

[ritwik06]

Homework Statement

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?

 

[Hurkyl]

Homework Statement

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?

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.

 

[GreenBeret]

Homework Statement

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?

that's not true always .

example :

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

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

 

[HallsofIvy]

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

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

 

[foxjwill]

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

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

why proof by induction? The way I was thinking of it is like this:
Rewriting each $$a_i$$ as $$1+c_i$$, we have

$$(1+c_1)+(1+c_2)+\cdots+(1+c_K)=K+c_1+c_2+\cdots+c_K=K+C \geq K$$​

 

[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.

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