# Thread: What is the best way to prove this conjecture?

1. ## What is the best way to prove this conjecture?

Okay, first post in the forums for me

This is my first year in high school geometry (i'm a freshman) and I'm loving it! I've run into a spot of trouble though...

My geometry teacher challenged us with a problem, and if I get it right, he will give me an A for the semester. I don't have the exact conjecture written down, but basically it is a formula to find the nth term in a sequence. For example, this formula could tell you what the 65th term in the sequence 4, 7, 13, 21.... (1st difference is 3, 6, 9, 2nd difference is 3,3,3). I have yet to learn proofs, which is why this is a challenge.

From the research I have done, I figure that mathematical induction would be the best method. I am guessing that if I can prove it works for a difference of 1, I can prove it for 2,3,4,5, etc. Am I correct in my thinking? Is there a better method? Also, any advice in the process; this is my first time proving a theorem (and he said it IS a theorem, as he has already proven it). Thanks in advance!

I'll edit this tomorrow when I write down the formula, sorry about its absence right now. If any more information is needed, just ask. I look forward to helping out in the algebra section, and spending some time on these forums!

2. If the second difference column is constant then the function is a quadratic.
That is,
$\displaystyle f(n)=an^{2}+bn+c.$
Use the first three numbers to solve for $\displaystyle a, b, c.$

3. "...and if I get it right, he will give me an A for the semester." Due to the philosophy of this website, I'll pass on this one.

4. Originally Posted by wonderboy1953
"...and if I get it right, he will give me an A for the semester." Due to the philosophy of this website, I'll pass on this one.
I expected this to come up, and I suppose I wasn't clear enough. I fully intend to work out the proof myself- once I know how to that is. My teacher said that we could do outside research (as we havn't learned proofs yet) and that is what I am doung. I don't want anyone to solve the problem for me, just explain the process of proving a conjecture using mathematical induction. Hope that clears some stuff up.

Okay, Here is all of the information, with a better explanation of what I need help on.

The theorem $\displaystyle A_k=[(k-b)*c]+a$ is used to solve for the nth term in an arithmetic sequence. The variables are as such:
$\displaystyle k=$the number term you are looking for (e.g. the 54th term or the 78th term)
$\displaystyle b=$the number of terms shown (e.g in the sequence of 2,4,6,8... this value would be 4)
$\displaystyle c=$the 1st difference
$\displaystyle a=$the last term shown (e.g. 8 in the above example)

A quick example in case I did not explain it thoroughly:
Given the sequence 3,6,9,12,15,18,21... determine the 86th term.

$\displaystyle A_k=[(k-b)*c]+a$

$\displaystyle A_k=[(86-7)*3]+21$

$\displaystyle A_k=[237]+21$

$\displaystyle A_k=258$

The 86th term is therefore 258. I understand all of the above. The challenge is to prove this as a theorem, and that is what I am confused about, as I have not yet learned proofs. It is my understanding that if I can prove it to be true when $\displaystyle c=0$, I can then prove it to be true for $\displaystyle n+1$. I don't quite understand how to do that, but I can figure it out with time. What I don't get is how to prove this for negative integers. Can anybody offer an explanation at this point? Thanks.

6. Your general idea about induction is correct. So, you prove the formula is correct for k = 1. Then you prove, under the assumption the formula for $\displaystyle A_k$ is correct, that the formula for $\displaystyle A_{k+1}$ is correct. One addition: you may have to do this for variables other than k. (Or maybe not. You should do something to deserve your A, don't you think?)

7. I leave my comment purposely obscure because of the whole A incentive.

If it were me I would replace a with something else.

8. Ah okay so my thinking is correct. Thats all I needed, really, to get started. Thanks again for all the help here. Quick question; can I mark this thread as 'solved', even if it truely isn't, as I'm not asking for any more help.

9. Originally Posted by fireballs619
Ah okay so my thinking is correct. Thats all I needed, really, to get started. Thanks again for all the help here. Quick question; can I mark this thread as 'solved', even if it truely isn't, as I'm not asking for any more help.
I see no problem with marking the thread solved; your main problem was "how to get started" and that has been solved.