Look, I just want someone to point me in the right direction. Like in the first problem, there are 4 different values for x because x ≠ 1, so there's more than one value for the first sum. Therefore I think I'm doing something wrong and I don't know how to start.Is this a joke? You want a helper on a math help forum to do your work for you? I assure you that no helper here would think that would be helpful to you. Read the posting guidelines, post back following those guidelines and get the best help available.
Jomo was (rudely) asking you to tell us what help you need, by either showing your thoughts or asking a specific question. You've now more or less done that.Look, I just want someone to point me in the right direction. Like in the first problem, there are 4 different values for x because x ≠ 1, so there's more than one value for the first sum. Therefore I think I'm doing something wrong and I don't know how to start.
How can you say that. You didn't even write anything. You just posted an image. From an image you want me to assume that you're asking for a hint. I was born at night, just not last night.Look, I just want someone to point me in the right direction. Like in the first problem, there are 4 different values for x because x ≠ 1, so there's more than one value for the first sum. Therefore I think I'm doing something wrong and I don't know how to start.
How do you know that just because you plug in different x-values that there will be more than one sum?Look, I just want someone to point me in the right direction. Like in the first problem, there are 4 different values for x because x ≠ 1, so there's more than one value for the first sum. Therefore I think I'm doing something wrong and I don't know how to start.
x4 = 1 - (1- x4)Hey thanks for the reply. But in the third to last one why did you do 1 - every fraction?
yeah i admit I wasn't very specific with my doubts sorry. And thanks for the other suggestion i will think about it.Jomo was (rudely) asking you to tell us what help you need, by either showing your thoughts or asking a specific question. You've now more or less done that.
Lex has offered an idea (in fact, a full solution to the first part). Here's another suggestion:
First, the fact that x might be one of 4 different values doesn't necessarily imply that the sum has more than one value! In fact, the question hints that there might be something interesting to investigate about the sum. (When a problem looks wrong, maybe it's just amazing. Trust the author and give it a try.)
Playing with various ideas, I wondered if any pairs of terms might add up to a constant. Think about adding the first and last term, and then the two middle terms. What happens in each case?
Thanks a lot you really helped@X049
What you say is correct for the second sum, so I wonder have they made a mistake with their indices in the second one.
They may mean [MATH]x^{2021}=1[/MATH] and mean to finish with [MATH]\frac{1}{1+x^{2020}}[/MATH], to mirror the first question.
In any case, there is one answer for all 5 roots in the first part (but you can find it trivially for x=1).
Looking at the first one, you should be able to complete the next one.
[MATH]\Sigma=\frac{1}{1+x}+\frac{1}{1+x^2}+\frac{1}{1+x^3}+\frac{1}{1+x^4}[/MATH]
Now [MATH]x^5=1[/MATH], so everywhere you see a 1, write [MATH]x^5[/MATH]
[MATH]\Sigma=\frac{x^5}{x^5+x}+\frac{x^5}{x^5+x^2}+\frac{x^5}{x^5+x^3}+\frac{x^5}{x^5+x^4}[/MATH]
[MATH]\Sigma=\frac{x^4}{1+x^4}+\frac{x^3}{1+x^3}+\frac{x^2}{1+x^2}+\frac{x}{1+x}[/MATH]
[MATH]\Sigma=1-\frac{1}{1+x^4}+1-\frac{1}{1+x^3}+1-\frac{1}{1+x^2}+1-\frac{1}{1+x}[/MATH]
[MATH]\Sigma=4-\Sigma[/MATH]
[MATH]\therefore \Sigma=2[/MATH]