From direct calculation, the answer is 527, but how to do algebraically?
x^5 + 1/x^5 = 2525
Let x + 1/x = u
x^2 + 2x(1/x) + 1/x^2 = u^2
x^2 + 1/x^2 = u^2 - 2
x^3 + 3x + 3/x + 1/x^3 = u^3
x^3 + 1/x^3 = u^3 - 3u
x^4 + 4x^2 + 6 + 4/x^2 + 1/x^4 = u^4
x^4 + 1/x^4 = u^4 - 4(u^2 - 2) - 6
x^4 + 1/x^4 = u^4 - 4u^2 + 2
x^5 + 5x^3 + 10x + 10/x + 5/x^3 + 1/x^5 = u^4
x^5 + 1/x^5 = u^5 - 5(u^3 - 3u) - 10u
x^5 + 1/x^5 = u^5 - 5u^3 + 5u = 2525
u^5 - 5u^3 + 5u - 2525 = 0
u=5 is a root of this equation
x^4 + 1/x^4 = u^4 - 4u^2 + 2
x^4 + 1/x^4 = 5^4 - 4*5^2 + 2 = 527
|
|
Posted by Larry
on 2024-10-26 08:58:39 |
Solution
