Could someone explain how this works? Pick 2 integers w/ difference greater than 1...

[Help please]

I was just thinking to myself and found this unusual pattern but cannot explain how it works

1. Pick 2 integers with a difference greater than 1 ( it will still work but the pattern isn't as noticeable)
2. Work out the number of integers between the 2 that you have chosen ( find difference and add 1) keep this number in mind
3. Multiply the 2 integers that you have chosen keep this answer in mind
4. Add 1 to the smaller integer that you have previously chosen and take 1 away from the larger
5. Multiply the 2 new numbers
6. You should find that the difference between the two products have a difference of the number that you have found in step 2.

Hopefully this all makes sense. Just ask if you don't understand. Thanks!

P.s. not sure if there is already a thread about this. Wasn't too sure how to put this pattern into words

 

[lev888]

First product: nm.
Second product: (n+1)(m-1)

What's their difference?
And double check your calculation of the number of integers between n and m.

 

[Help please]

Thanks! Makes much more sense now. mn+m-n-1. The end is just the number of integers between added. Yeah your right it was take away 1 not add one

 

[JeffM]

I was just thinking to myself and found this unusual pattern but cannot explain how it works

1. Pick 2 integers with a difference greater than 1 ( it will still work but the pattern isn't as noticeable)
2. Work out the number of integers between the 2 that you have chosen ( find difference and add 1) keep this number in mind
3. Multiply the 2 integers that you have chosen keep this answer in mind
4. Add 1 to the smaller integer that you have previously chosen and take 1 away from the larger
5. Multiply the 2 new numbers
6. You should find that the difference between the two products have a difference of the number that you have found in step 2.

Hopefully this all makes sense. Just ask if you don't understand. Thanks!

P.s. not sure if there is already a thread about this. Wasn't too sure how to put this pattern into words

Don't put it into words; put it into math.

\(\displaystyle \text {STEP I: } m,\ n \in \mathbb Z \text { and } m - n \ge 2 \implies m > n.\)

Your step 2 is wrong, probably a typo. The number of integers between 8 and 11 is two, namely 9 and 10,
which is not equal to 11 - 8 + 1 = 4. It is equal to 11 - 8 - 1 = 2.

\(\displaystyle \text {STEP II: } x = m - n - 1.\)

\(\displaystyle \text {STEP III: } y = m * n = mn.\)

\(\displaystyle \text {STEP IV: } z = (m - 1)(n + 1) = mn + m - n - 1.\)

\(\displaystyle \text {STEP V: } x = z - y \ \because \ (mn + m - n - 1) - mn = m - n - 1.\)