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

In summary: Proof by induction: assume that for all i, 1+c_i>K+c_1+c_2+\cdots+c_K.now, let's prove that for all i, 1+c_i>K+c_1+c_2+\cdots+c_K if and only if c_i is the smallest integer such that 1+c_i>K+c_1+c_2+\cdots+c_K.for all i such that c_i<K, we have 1+c_i>K+c_1+c_2+\cdots+c_K.thus
  • #1
ritwik06
580
0

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:

[tex]a_{1}+a_{2}+a_{3}+....a_{K}>K[/tex]
[tex]a_{i}[/tex] is a positive integer.
I know this is always true. but how should I prove it mathematically?
 
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  • #2
ritwik06 said:

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:

[tex]a_{1}+a_{2}+a_{3}+....a_{K}>K[/tex]
[tex]a_{i}[/tex] 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.
 
  • #3
ritwik06 said:

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:

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

that's not true always .

example :

take [tex] a_i=1[/tex] , for all [tex]i[/tex].

So , we get [tex]\overbrace{1+1+\cdots+1}^K =K \not > K [/tex]
 
  • #4
I presume you mean [itex]a_1+ a_2+ \cdot\cdot\cdot+ a_K\ge K[/itex] where the an are positive integers.

Looks like a good candidate for "proof by induction".
 
  • #5
HallsofIvy said:
I presume you mean [itex]a_1+ a_2+ \cdot\cdot\cdot+ a_K\ge K[/itex] where the an 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 [tex]a_i[/tex] as [tex]1+c_i[/tex], we have
[tex](1+c_1)+(1+c_2)+\cdots+(1+c_K)=K+c_1+c_2+\cdots+c_K=K+C \geq K[/tex]​
 
  • #6
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
 

1. How do I approach a proof for a_1 + a_2 + ... + a_K > K?

To start, it is important to understand the definitions and properties of the terms involved. This includes understanding what a_1, a_2,... a_K represent and how they relate to each other. From there, you can try different strategies such as direct proof, contradiction, or induction to prove the inequality.

2. What are the common mistakes to avoid in a proof for a_1 + a_2 + ... + a_K > K?

One common mistake is assuming that the given statement is true without any proof. It is important to use the definitions and properties of the terms involved to logically and rigorously prove the inequality. Another mistake is using circular reasoning, where the conclusion is used to prove the premise. It is also important to be careful with algebraic manipulations and to clearly state each step of the proof.

3. How can I break down a complex statement like a_1 + a_2 + ... + a_K > K into simpler parts for the proof?

One approach is to use the properties of inequalities to break down the statement into smaller inequalities. For example, you can show that a_1 > 1, a_2 > 1,... a_K > 1 and then use the property that the sum of positive numbers is always greater than the number of terms to prove the original statement.

4. Can I use examples or counterexamples to prove a_1 + a_2 + ... + a_K > K?

No, examples or counterexamples are not sufficient to prove a statement in mathematics. A proof must be logical and valid for all possible cases. However, you can use examples to gain intuition and guide your approach for the proof.

5. What should I do if I am stuck on a proof for a_1 + a_2 + ... + a_K > K?

If you are stuck, it can be helpful to take a break and come back to the problem later with a fresh perspective. You can also consult with a classmate or instructor for guidance. Additionally, looking at similar proofs or consulting with other resources can provide insight and inspiration for your own proof.

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