# Thread: simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

1. ## simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

I would love to show ANY effort of my own, but it seams to me that I should use some theorem (of which I have never heard of) in order to attack this problem.

2. ## Re: simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

hello there notice that u can seperate this sequence
into two sequences which are both geometric
the first is:x+x^2+x^3+...........+x^n
the second is:1/x +1/x^2+1/x^3+.........1/x^n
now just use the formula for the sum of ageometric sequence

3. ## Re: simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

Originally Posted by islam
hello there notice that u can seperate this sequence into two sequences which are both geometric
the first is:x+x^2+x^3+...........+x^n
the second is:1/x +1/x^2+1/x^3+.........1/x^n
now just use the formula for the sum of ageometric sequence
That is not the case.
For example: $\displaystyle \left( {x^3 + \frac{1}{{x^3 }}} \right)^3 = x^9 + 3x^3 + 3x^{ - 3} + x^{ - 9}$

4. ## Re: simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

sorry very sorry
i just missed the exponent

5. ## Re: simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

I'm starting to think the whole problem was a typo, and the real form should be:
(x+1/x) + (x^2 + 1/x^2) +...+(x^n + 1/x^n)
so it's quite simple.

But thanks for the help!

6. ## Re: simplify: (x+1/x) + (x^2 + 1/x^2)^2 +...+(x^n + 1/x^n)^n

Originally Posted by witek123
I'm starting to think the whole problem was a typo, and the real form should be:
(x+1/x) + (x^2 + 1/x^2) +...+(x^n + 1/x^n)
so it's quite simple.
I agree with that.