Differences of squares
Subtract level by level until the numbers match.
Subtract, then subtract again
Take the squares 1, 4, 9, 16, 25, 36, 49. The differences between neighbours are the odd numbers 3, 5, 7, 9, 11, 13. Now take the differences of those numbers.
What will the differences of 3, 5, 7, 9, 11, 13 look like?
What this lesson covers
The idea
When differences of consecutive perfect squares are computed successively, after two levels all the differences are the same.
Subtract, then subtract again
Take the squares 1, 4, 9, 16, 25, 36, 49. The differences between neighbours are the odd numbers 3, 5, 7, 9, 11, 13. Now take the differences of those numbers.
What will the differences of 3, 5, 7, 9, 11, 13 look like?
- They keep changing
- They are all the same
- We get the squares again
Find the differences
Press Find differences to subtract neighbours. Do it twice. Then change the first square with − and +: does the pattern change?
Two levels, all the same
When the differences of consecutive perfect squares are computed successively, after two levels all the differences are the same.
Why? The first differences are odd numbers, and odd numbers go up by 2 each time (3, 5, 7, 9, …). So the differences of the differences are all 2.
Notes
When the differences of consecutive perfect squares are computed successively, after two levels all the differences are the same.
Check yourself
The squares 10², 11², 12², 13² are 100, 121, 144, 169. Find the differences, then the differences of those. What number do you get at the second level?
Answer: 2
First level: 21, 23, 25. Second level: 23 − 21 = 2 and 25 − 23 = 2. All the same!
After how many levels of differences are all the numbers the same, for consecutive perfect squares?
The first differences of consecutive squares begin 21, 23, 25, … What is the next first-level difference?
Answer: 27
The differences go up by 2 each time, so after 25 comes 27.
The squares go 36, 49, 64, … and their differences are 13, 15, …. Use the pattern to find the next square.
Answer: 81
The next difference is 15 + 2 = 17, and 64 + 17 = 81 = 9².
- One level. After one level we get odd numbers such as 3, 5, 7, 9. They are not all the same.
- Two levels — correct. Yes! After two levels every difference is 2.
- Three levels. They are already all the same after two levels.
- Never. They do match: after two levels all the differences are 2.