Difference analysis for odd positive integers

1,3,5,7,9,11

1st Level 2 2 2 2 2

X=1, X= 1

X=2, X=3

X=3, X=5

ax^2+bx+c = 0
a(1)^2+b(1)+c = 1
a(2)^2+b(2)+c = 3
a(3)^2+b(3)+c = 5

Subtract One parabola from the next to get the values for a, b and c.
a+b+c = 1
- 4a+2b+c = 3
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-3a-1b = -2, 3a+b = 2

4a+2b+c = 3
- 9a+3b+c = 5
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- 5a-b+c = -2, 5a+b=2

Simplify for a
3a+b = 2
-5a+b = 2
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-2a = 0, a = 0

Now that you have a simplify for b
3(0)+b = 2
b = 2

Solve for c
a+b+c = 1
(0) + (2) +c = 1
c = -1

Create the formula
2x -1= 0
Assume that n = x, so Tn = 2n-1

Plug in for the arithmetic series equation
Sn = n/2(T1+Tn)
Sn = n/2(T1+2n-1)
Sn=n^2