What is the best way to prove this conjecture?

Printable View

• Aug 31st 2010, 04:08 PM
fireballs619
What is the best way to prove this conjecture?
Okay, first post in the forums for me(Nod)

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!
• Sep 1st 2010, 01:26 AM
BobP
If the second difference column is constant then the function is a quadratic.
That is,
$f(n)=an^{2}+bn+c.$
Use the first three numbers to solve for $a, b, c.$
• Sep 1st 2010, 10:35 AM
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.
• Sep 1st 2010, 12:47 PM
fireballs619
Quote:

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.
• Sep 2nd 2010, 04:47 PM
fireballs619
More information
Okay, Here is all of the information, with a better explanation of what I need help on.

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

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

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

$A_k=[237]+21$

$A_k=258$

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 $c=0$, I can then prove it to be true for $n+1$. 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.
• Sep 2nd 2010, 09:39 PM
Pim
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 $A_k$ is correct, that the formula for $A_{k+1}$ 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?)
• Sep 2nd 2010, 10:29 PM
undefined
I leave my comment purposely obscure because of the whole A incentive.

If it were me I would replace a with something else.
• Sep 3rd 2010, 03:33 AM
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.
• Sep 3rd 2010, 06:38 AM
undefined
Quote:

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. (Happy)