Proving a Conjecture by Induction

In summary, the conversation discusses proving a conjecture by induction and the series (n^2) + n = (1/3) ((n^3)+3(n^2)+2n). The speaker doubts the validity of the statement and suggests that the correct statement may be \sum_{k=1}^{n} \left(k^2 + k\right) = \frac{1}{3}(n^3+3n^2+2n). They also provide a proof for this statement using induction. Towards the end, there is a discussion about interpreting a question and the calculation of 1^2 + 2^2 + 3^2...+n^2. The suggested solution is to
  • #1
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I am supposed to prove a conjecture by induction. I have worked out that the series can be described by:

a = 2,6,12,20,30
S = 2+8+20+40...

(n^2) + n = (1/3) ((n^3)+3(n^2)+2n)

However, i cannot prove it by induction. It seems like there is smth wrong with the K^3.
 
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  • #2
What series? And what precisely is it that you want to prove? I doubt that you actually want to prove that (n^2) + n = (1/3) ((n^3)+3(n^2)+2n), because that's not true (for all n).
 
  • #3
This should help

Muzza said:
What series? And what precisely is it that you want to prove? I doubt that you actually want to prove that (n^2) + n = (1/3) ((n^3)+3(n^2)+2n), because that's not true (for all n).
I think Link means [tex]\sum_{k=1}^{n} \left(k^2 + k\right) = \frac{1}{3}(n^3+3n^2+2n)[/tex], which is in fact true.
The inductive proof goes like this:
i. [tex]\sum_{k=1}^{n} \left(k^2 + k\right) = \frac{1}{3}(n^3+3n^2+2n)[/tex] , holds for [tex]n=1[/tex];
ii. Assume [tex]\sum_{k=1}^{n} \left(k^2 + k\right) = \frac{1}{3}(n^3+3n^2+2n)[/tex] holds for some fixed n, so that
[tex]\sum_{k=1}^{n} \left(k^2 + k\right) = \frac{1}{3}(n^3+3n^2+2n)\Rightarrow \sum_{k=1}^{n} \left(k^2 + k\right) + \left((n+1)^2 + (n+1)\right) = \frac{1}{3}(n^3+3n^2+2n) + \left((n+1)^2 + (n+1)\right)[/tex]
[tex]\Rightarrow \sum_{k=1}^{n+1} \left(k^2 + k\right) = \frac{1}{3}(n^3+6n^2+11n+6) = \frac{1}{3}\left((n+1)^3+3(n+1)^2+2(n+1)\right) [/tex].
Therefore, [tex]\sum_{k=1}^{n} \left(k^2 + k\right) = \frac{1}{3}(n^3+3n^2+2n)[/tex] holds for every positive integer n.
 
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  • #4
thanks! but there is another problem arising from this, regarding the interpretation of a question.

Based on the results obtained above, how can i calculate 1^2 + 2^2 + 3^2...+n^2?

the problem is the word "calculate", do you think they want me to use the same method to derive a general expression for this series, or actually get an numerical value (i don't think its obtainable)? I am the only one in class doing this investigation and I cannot reach my teacher this week so advice on the meaning of this is appreciated.
 
  • #5
Well, you now know that
[tex]\sum_{k=1}^{n} \left(k^2 + k\right)=\sum_{k=1}^nk^2+ \sum_{k=1}^nk = \frac{1}{3}(n^3+3n^2+2n)[/tex]
If you know a formula for
[tex]\sum_{k=1}^{n} k[/tex]
just subtract that from both sides.
 

What is a conjecture?

A conjecture is a statement or proposition that is believed to be true, but has not yet been proven or disproven. It is often based on observation or intuition, but requires evidence and logical reasoning to be accepted as true.

What is induction?

Induction is a mathematical proof technique that involves proving a statement for a specific base case and then showing that if the statement is true for any given case, it must also be true for the next case. This process continues until the statement is proven to be true for all cases.

How do you use induction to prove a conjecture?

To prove a conjecture by induction, you must first establish the base case by showing that the statement is true for some initial value. Then, you must assume that the statement is true for an arbitrary case and use this assumption to show that the statement is also true for the next case. By repeating this process, you can show that the statement is true for all cases and therefore, the conjecture is proven.

What are the steps in an induction proof?

The steps in an induction proof include: stating the base case, assuming the statement is true for an arbitrary case, using this assumption to prove the statement for the next case, and repeating until the statement is proven for all cases. Finally, you must conclude that the conjecture is true based on the proof.

What are some common challenges in proving a conjecture by induction?

Some common challenges in proving a conjecture by induction include: finding the correct base case, determining the correct pattern to use in the proof, and ensuring that the proof is logically sound and covers all possible cases. It is also important to clearly explain each step of the proof and justify why it is necessary.

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