Question about Induction Hypothesis

  • Thread starter flyingpig
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In summary: That's just a fancy name for "assume the statement is true for n = k". You need to make that assumption in order to get off the ground. So you should write something like "Assume that the statement is true for n = k." Then, you use that assumption to show that the statement is also true for n = k + 1. Once you show that, you have completed the proof.
  • #1
flyingpig
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Homework Statement

Prove that for all integers [tex]n \geq 1[/tex], one has

[tex]1 + 2 + ... + n = \frac{n(n+1)}{2}[/tex]

(1) S(1) = 1, true

(2) Let n = k + 1

[tex]1 + 2 + ... + k + (k + 1) = \frac{(k+`1)(k + 2)}{2}[/tex]

The Attempt at a Solution



Why is the last series

[tex]1 + 2 + ... + k + (k +1)[/tex] instead of [tex]1 + 2 +...+ (k + 1)[/tex]?
 
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  • #2
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
flyingpig said:

Homework Statement




Prove that for all integers [tex]n \geq 1[/tex], one has

[tex]1 + 2 + ... + n = \frac{n(n+1)}{2}[/tex]

(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.
flyingpig said:
[tex]1 + 2 + ... + k + (k + 1) = \frac{(k+`1)(k + 2)}{2}[/tex]





The Attempt at a Solution



Why is the last series

[tex]1 + 2 + ... + k + (k +1)[/tex] instead of [tex]1 + 2 +...+ (k + 1)[/tex]?

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
Is it bad that I don't show it?
 
  • #5
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.
 

1. What is a proof and why is it important?

A proof is a logical and systematic demonstration that a statement is true based on established axioms and previously proven theorems. It is important because it allows scientists to validate their theories and conclusions, ensuring that they are supported by evidence and reasoning.

2. How do you construct a valid proof?

To construct a valid proof, one must start with a clear and precise statement of the theorem or claim to be proven. Then, using established axioms and previous theorems, the proof must proceed in a logical and step-by-step manner, providing evidence and reasoning for each step. The proof must also be free of errors and gaps in logic.

3. What makes a proof convincing?

A convincing proof is one that is constructed using sound logic and evidence, and is easily understandable to others. It should also be free of any errors or gaps in reasoning, and should align with the established axioms and theorems of the relevant field of study.

4. How do you know when a proof is complete?

A proof is considered complete when it has reached the conclusion or solution that was set out to be proven, and all steps have been logically connected and supported by evidence. It should also be free of any errors or gaps in reasoning.

5. Can a proof be proven wrong?

Yes, a proof can be proven wrong if it is found to have errors or gaps in reasoning, or if new evidence or counterexamples are discovered that contradict the conclusion. This is why it is important for proofs to be thoroughly reviewed and critiqued by other scientists in the field before being accepted as valid.

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