Sum of odd numbers and square roots

Do you know that:

Sum - odd numbers
1=1
1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
....

Square roots
The square root of 1 is 1
The square root of 4 is 2
The square root of 9 is 3
The square root of 16 is 4
The square root of 25 is 5
...

Very interesting! 😏

Originally posted by Milou
Do you know that:

Sum - odd numbers
1=1
1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
....

Square roots
The square root of 1 is 1
The square root of 4 is 2
The square root of 9 is 3
The square root of 16 is 4
The square root of 25 is 5
...

Very interesting! 😏

sum_odd(1..n) = average value * number of entries
= (1 + n)/2 * (1 + n)/2
=((1 + n)/2)^2

2^2=4
3^2=(2+1)(2+1)=FOIL=4+2+2+1=9
(X+1)^2=X*X+X+X+1
(X+1)^2=X(X+2)+1

1+3=4
1+3=1+1+1+1
X=1 X+1=2 2=number of odd numbers being added
1+3=X+X+X+1
1+3=3X+1
3X=1+1+1
1+1+1=1(1+2) because X=1, 1(1+2)=X(X+2) thus
1+3=X(X+2)+1
1+3+5=?
X=2 (the old X+1) X+1=3 3=the number" "added
1+3+5=X(X+2)+1=3(3+2)+1=

i have no idea where this is going.