Can any number of the form [math]\frac{4\cdot 10^{2n-1}-31}{9}[/math] ever be a perfect square? This statements needs to be mathematically proven (in detail!).
My apologies. The initial question was if any number which consists an odd amount of consecutive digits of 4 and ends in a 1 (4441, 444441, and so on) can be a perfect square. So far I have found a general formula to construct such a number (as shown above) and thought about a proof using induction, which I am currently stuck at.Okay. What has been covered recently in class? What are your thoughts? What have you tried? How far have you gotten?
Please be complete. Thank you!
I would probably start by plugging in some values for n, just to get a sense of how the expression works. I'd also check the definition of "perfect square" in your context.Can any number of the form [math]\frac{4\cdot 10^{2n-1}-31}{9}[/math] ever be a perfect square? This statements needs to be mathematically proven (in detail!).
I actually came up with that expression as the question was initially different (my mistake I should have said that in the question). It's for number like 4441, 444441, 44444441 (odd amount of 4 followed by 1). I have thought about using induction as a proof since I believe it's fine for existence or non-existenceI would probably start by plugging in some values for n, just to get a sense of how the expression works. I'd also check the definition of "perfect square" in your context.
Then I might think about what it would take to prove this "can it ever" (that is, existence) claim, one way or the other (existence or non-existence).
I actually came up with that expression as the question was initially different (my mistake I should have said that in the question). It's for number like 4441, 444441, 44444441 (odd amount of 4 followed by 1). I have thought about using induction as a proof since I believe it's fine for existence or non-existence
Please (1) show us the actual original problem as given to you; and (2) show us your attempt at induction.I'm terribly sorry, It seems like I have made a typo in my original post.
Here is the updated term[math]\frac{4*10^{2n}-31}{9}[/math]
My mistake, certainly. The problem was this: "For any given number of the form 44...41, which consists of an odd number of digits of 4, followed by a digit of 1, could this number ever be a perfect square?". Perfect square meaning in this case, can the square root of such a number be a positive integer.Please (1) show us the actual original problem as given to you; and (2) show us your attempt at induction.
Proofs of existence and non-existence are very different. For existence, you just need to show an example; no induction is needed. So I suppose you must be trying to show non-existence?
(For your initial post, I found that n=1 gave an example.)
Sure, 81 for example or 121For any given number of the form 44...41 could this number ever be a perfect square?
They just want to know if there is at least one number of this form. So far you haven't found any. So either you find 1 or proof that there are none.
Can a perfect square even end in a 1? I would settle that before I look any further! How about 41?
Could you comment on this approach?The answer is 'no', and the proof (by contradiction) is pretty straightforward.
No, your logic is not correct. The problem is not asking you to show that every number of the form 44....41 is a perfect square. They are asking if ANY number of the form 44...41 is a perfect square.Could you comment on this approach?
Suppose k is an integer, where
[math]9k^2=4\cdot10^{2n}-31[/math]With this, the right side is always odd, since 4*10^{2n} is even and subtracting 31 makes it odd. However the left side could be even (e.g 9*2^2 = 36). This is a contradiction, so the statement can not be true.
If such a number exist, but NOT for all n, then you can't use induction. You can only use induction to show that such a number doesn't exist. That is all numbers of the form 44...41 are NOT perfect squares.I actually came up with that expression as the question was initially different (my mistake I should have said that in the question). It's for number like 4441, 444441, 44444441 (odd amount of 4 followed by 1). I have thought about using induction as a proof since I believe it's fine for existence or non-existence
I see, thank you for pointing this out. Since any odd k does makes both sides odd, the parity argument doesn't work. I have though about other arguments like checking the modulo on both sides but they seem to be always the same when k is odd. Could you think of any other way of showing a contradiction when k is odd, because I can't seem to figure one out?No, your logic is not correct. The problem is not asking you to show that every number of the form 44....41 is a perfect square. They are asking if ANY number of the form 44...41 is a perfect square.
You clearly showed in you example that if k is even that the lhs and the rhs can't be equal. Wonderful!! Now show that the two sides are never equal when k is odd or show an example of a number in the form 44...41 being a perfect square!
For the record, you already showed an example of a number in the form 44...41 that was not a perfect square. Then you go again and show us an infinite amount of numbers in the form 44...41 that are not perfect squares. You keep thinking that is enough. If it was enough then you solved this problem twice. Unfortunately, what you showed is not enough. Again, now show that when k is odd that ....
81, 121, ...Can a perfect square even end in a 1?
Actually, I did not use induction.You were told in a previous post by blamocur that there is no such number!
Therefore you should try to prove this by induction.
Yes, but the poster needs to 1st really that ending in 1 is even possible! Perfect squares can end in 0, 1, 4, 5, 6 or 9. I realized that on my own while in JHS.81, 121, ...