A formula for the next power of 5?

Question

I have come up with the following formulas:

To the next square:
x^2 + (x - (x - 1)) + 2x = (x + 1)^2

To the next cube:
x^3 + (x^3 - (x - 1)^3) + 6x = (x + 1)^3

To the next fourth:
x^4 + (x^4 - (x - 1)^4) + (12(x^2) + 2) = (x + 1)^4

I cannot figure out a formula for the power of 5 sequence. Any ideas are welcome. Thanks

Answer 1

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This is the Pascal triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
...
Basically each number in the next row is the sum of 2 numbers above is. This is the basis for calculating en exponent of a sum

Based on the triangle, we can prove that:

(x+y)^(n-1) = {a[1]}x^n + {a[2]}*x^(n-1)*y + {a[2]}*x^(n-2)*y^2 + ... + {a[n-2]}*x^2*y^(n-2) + {a[n-1]}*x*y^(n-1) + {a[n]}y^n

n represents the position of the row in the triangle and the number in [ ] represents the position of the number in that row

For example:
(x+y)^0 = 1
(x+y)^1 = x + y
(x+y)^2 = x^2 + 2xy + y^2
(x+y)^3 = x^3 + 3*x^2*y + 3*x*y^2 + y^3
(x+y)^4 = x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + y^4
(x+y)^5 = x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5
...
Replace x, y with any values

(x+1)^5 = x^5 + 5*x^4 + 10*x^3 + 10*x^2 + 5*x + 1

Simplify it whichever way you like, but it'll eventually end up like this.