jmc

Question

x is a real number with the property that x+1x = 3. Let S_m = x^m + 1x^m. Determine the value of S_7.

Accepted Answer

We can calculatex^2 + 1x^2 = (x + 1x)^2 - 2 = 3^2 -2 = 7.Similarly,x^3 + 1x^3 = (x + 1x) (x^2 + 1x^2) - (x + 1x) = 3 7 - 3 = 18andx^4 + 1x^4 = (x^2 + 1x^2)^2 - 2 = 7^2 - 2 = 47.Finally,x^7 + 1x^7 = (x^3 + 1x^3) (x^4 + 1x^4) - (x + 1x) = 18 47 - 3 = 843.