+170 Let \(r\) be the positive real solution to \(x^3 + \frac{2}{5} x - 1 = 0. \) Find the exact numerical value of \(r^2 + 2r^5 + 3r^8 + 4r^{11} + \dotsb.\)
Solve for x:
x^3 + (2 x)/5 - 1 = 0
Bring x^3 + (2 x)/5 - 1 together using the common denominator 5:
1/5 (5 x^3 + 2 x - 5) = 0
Multiply both sides by 5:
5 x^3 + 2 x - 5 = 0
Divide both sides by 5:
x^3 + (2 x)/5 - 1 = 0
Change coordinates by substituting x = y + λ/y, where λ is a constant value that will be determined later:
-1 + 2/5 (y + λ/y) + (y + λ/y)^3 = 0
Multiply both sides by y^3 and collect in terms of y:
y^6 + y^4 (3 λ + 2/5) - y^3 + y^2 (3 λ^2 + (2 λ)/5) + λ^3 = 0
Substitute λ = -2/15 and then z = y^3, yielding a quadratic equation in the variable z:
z^2 - z - 8/3375 = 0
Find the positive solution to the quadratic equation:
z = 1/450 (225 + sqrt(51105))
Substitute back for z = y^3:
y^3 = 1/450 (225 + sqrt(51105))
Taking cube roots gives (225 + sqrt(51105))^(1/3)/(2^(1/3) 15^(2/3)) times the third roots of unity:
y = (225 + sqrt(51105))^(1/3)/(2^(1/3) 15^(2/3)) or y = -(-225 - sqrt(51105))^(1/3)/(2^(1/3) 15^(2/3)) or y = ((-1)^(2/3) (225 + sqrt(51105))^(1/3))/(2^(1/3) 15^(2/3))
Substitute each value of y into x = y - 2/(15 y):
x = ((sqrt(51105) + 225)/2)^(1/3)/15^(2/3) - 2 (2/(15 (sqrt(51105) + 225)))^(1/3) or x = -((-sqrt(51105) - 225)/2)^(1/3)/15^(2/3) - 2 (-1)^(2/3) (2/(15 (sqrt(51105) + 225)))^(1/3) or x = 2 ((-2)/(15 (sqrt(51105) + 225)))^(1/3) + ((-1)^(2/3) ((sqrt(51105) + 225)/2)^(1/3))/15^(2/3)
Bring each solution to a common denominator and simplify:
x≈0.867559241390112
x≈-0.433779620695056 - 0.982086695761143 i
x≈-0.433779620695056 + 0.982086695761143 i
Use the Real Value to sum up the sequence:
sumfor(r, 1, 10000, (r*0.867559^(3*r-1))) = 6 1/4
