An integer ?≥2n≥2 is called square-positive

[argon]

An integer ?≥2n≥2 is called square-positive if there are ? consecutive positive integers whose sum is a square. Determine the first four square-positive integers.

So I have found the first four square-positive numbers, but I need to prove that why it 44 is not a square-positive number and I also need to write a general formula for determining whether a number is square-positive or not. I have tried to write the sum of consecutive positive integers like this ?+?+1+?+2+?+3....?−1 and I wrote it like this for all numbers, and part of the proof for why 44 isn't a square-positive number is that 4?+64a+6 is not divisible with 44. But I haven't got so far.

Here is my answer:
2 : 4 + 5 = 9 which is 3^2
3 : 2 + 3 + 4 = 9 which is 3^2
5: 18 + 19 + 20 + 21 + 22 = 100 which is 10^2
6: 35 + 36 + 37 + 38 + 39 + 40 = 225 which is 15^2

Interesting fact is that for all odd numbers and some even numbers like 6 and 10, you can find out which number is the first (the one you start with and then go forward here like 3, 2, 18 and 35) using this formula :
(I show it in an example because I still can't write it algebraically):

For example: the sum of 95 subsequent numbers is 5n + 10
(10^2 - 10) /5 = 18
So your first number is 18
And if you keep adding, 18 + 19 + 20 + 21 + 22 you get 100 which is 10^2, the same number you squared.

 

[Steven G]

?≥2n≥2 tells me that ?≥2n and that n≥2.

If n≥2 how can ?≥2n? After all if ?≥2, ie n>0, then ?≥2n implies that 1≥2.
I did not read past that. Either I am missing something or you are starting off with something wrong.

 

[argon]

Sorry I can't edit it now but it's just n≥2, I wrote it twice.

?≥2n≥2 tells me that ?≥2n and that n≥2.

If n≥2 how can ?≥2n? After all if ?≥2, ie n>0, then ?≥2n implies that 1≥2.
I did not read past that. Either I am missing something or you are starting off with something wrong.

 

[Dr.Peterson]

Actually, you didn't type it twice; you pasted from a source that includes math expressions in two formats, and you failed to proofread. That's also true with the number 4 (changed to 44), 4a+6 (changed to 4a+64a+6) and maybe something I've missed. (I don't know how "5" turned into "95"!) The decent thing would have been to repost the whole question, corrected, to help others out. I don't know why I so often seem to be the only one who recognizes this and can see past it.

I've corrected it here, I think:

An integer n≥2 is called square-positive if there are ? consecutive positive integers whose sum is a square. Determine the first four square-positive integers.

So I have found the first four square-positive numbers, but I need to prove that why it 4 is not a square-positive number and I also need to write a general formula for determining whether a number is square-positive or not. I have tried to write the sum of consecutive positive integers like this ?+?+1+?+2+?+3....?−1 and I wrote it like this for all numbers, and part of the proof for why 4 isn't a square-positive number is that 4a+6 is not divisible with 4. But I haven't got so far.

Here is my answer:
2 : 4 + 5 = 9 which is 3^2
3 : 2 + 3 + 4 = 9 which is 3^2
5: 18 + 19 + 20 + 21 + 22 = 100 which is 10^2
6: 35 + 36 + 37 + 38 + 39 + 40 = 225 which is 15^2

Interesting fact is that for all odd numbers and some even numbers like 6 and 10, you can find out which number is the first (the one you start with and then go forward here like 3, 2, 18 and 35) using this formula :
(I show it in an example because I still can't write it algebraically):

For example: the sum of 95 subsequent numbers is 5n + 10
(10^2 - 10) /5 = 18
So your first number is 18
And if you keep adding, 18 + 19 + 20 + 21 + 22 you get 100 which is 10^2, the same number you squared.

I think you're saying you worked out all this without ever writing a general formula, perhaps largely by trial and error? How do you choose 10 in your example for n=5?

Have you learned about the formula for the sum of an arithmetic series? Apply it to the sum of n consecutive integers starting at a.

But there are a couple things missing from what you've shown us. First, it appears that in addition to the problem itself in your first line, you have been given some additional instructions, including the part about 4a+6. Where does the problem come from? What is the entire question as given to you?

Second, we need to know what background you have, on the basis of which you are expected to do this. Can you tell us more of this context? (I've also seen that you either submitted this to Stack Exchange, or copied it from there.)