Hard to explain but like 1 squared is 1
2 is 4 that’s 3 difference
3 squared is 9 that’s a 5 difference
4 is 16 that’s a 7 difference. It just keeps rising by 2 why is this?
In: 5
Let n be any natural number. Then the square of n is n^2. The difference between n^2 and (n + 1)^2 is (n + 1)^2 – n^2 = (n + 1 + n)(n + 1 – n) = 2n + 1
(N+1)^2 – N^2 = N^2 + 2N + 1 – N^2 = 2N + 1
The difference between two consecutive squares in 2N + 1. It’s an odd number which increases by 2 each time.
You can try to visualize this. For example:
“`
ooooo
oooo xxxxo
ooo xxxo xxxxo
oo xxo xxxo xxxxo
xo xxo xxxo xxxxo
“`
If you think of Xs as the previous square and the Os as what you need to add to get the new square, you’ll see it’s the length of the old square plus the height of the old square plus 1. This is where the 2N+1 that others have mentioned comes from.
That 6×6 is 5×5 plus 5 and 6 is magnificent. I understand.
Can it work with anything? 15 squared is 225. By that logic 16 squared is 225 +15 +16 equals 256. HOLY HELL THANK YOU SO MUCH WHAT ABOUT 532 squared
That’s 283024 plus 532 plus 533 equals 284089
533 squared is just that.
YOU JUST.BLEW MY MIND. NEVER BEFORE HAS SQUARE NUMBERS BEEN EASIER. I’m shocked there’s this any rules for finding squared numbers.
The previous comment was deleted but thank you!
To go from 1^2 to 2^2 you add 3 (1+2)
To go from 2^2 to 3^2 you add 5 (2+3)
To go from 10^2 to 11^2 you add 21 (10+11)
See the pattern? You add both consecutive numbers to the square of the first to get the square of the second. In general: n^2 to (n+1)^2 you add n and n+1. To go from (n+1)^2 to (n+2)^2 add n+1 and n+2, so the increase will be 2 each time.
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