t^5 + 1/t^5 = (1 + 1/t)(t^4 - t^3 * 1/t + t^2 * 1/t^2 - t * 1/t^3 + 1/t^4)
= (t + 1/t)(t^4 + 1/t^4 -(t^2 +1/t^2) + 1)
Lets stop there,
Now we need to find t^4 + 1/t^4 and t^2 + 1/t^2
t^2 + 1/t^2 = (t + 1/t)^2 - t * 1/t
t^2 + 1/t^2 = 3^2 - 1
= 9 - 1 = 8
t^4 + 1/t^4:
(t^2 + 1/t^2)^2 = t^4 + 2 + 1/t^4
Soo, t^4 + 1/t^4 = 8^2 - 2
t^4 + 1/t^4 = 64 - 2 = 62
Now we can plug this back!
(t + 1/t)(62 - 8 + 1) = 3 * 55 = 165
I love these types of questions and I kinda just reviewed this yesterday!
I hope you understand!
