Sum of 2 Squares
The sum of two squares (a2 + b2) has a special formula:a2 + b2 = (a+b)2 − 2ab
Example: What is 162 + 42 ?
162 + 42 = (16+4)2 − 2×16×4
= (20)2 − 32×4
= 400 − 128
= 272
And this is why it works:
The 4 parts make the whole area: a2 + 2ab + b2 = (a+b)2
Subtract 2ab from both sides: a2 + b2 = (a+b)2 − 2ab
It is related to one of the Special Binomial Products.
Sum of a Sequence of Squares
We can sum a sequence of squares using a special formula from Partial Sums:
A shortcut when summing k2
Example: 12 + 22 + 32 + 42 + 52
Use the formula:
12 + 22 + 32 + 42 + 52 =
5(5+1)(2×5+1)
6
=
5×6×11
6
=55
See Partial Sums for more details.