Math Equation - Need help symplifying

[Krasz]

Hey guys i have been attempting this problem for days. If you can help me with it and explain how to solve it, I would be extremely grateful. I wrote my answer on the right. Thanks!

 

[2^Oscar]

The image isn't loading for me - could you possibly just type the equation in or is it too complex?


Oscar

 

[Krasz]

Ill give it a whirl.

1+1/x+1/x+1/x+1/x+1

Now, the second number in the equations is always the one being divided by x + 1.
It is not both numbers that are divided, just the second. This series goes all the way down.

 

[statdad]

So is it like this?

$$1+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+1}}}}$$

 

[arildno]

Start with the bottom:
$$x+\frac{1}{x+1}=\frac{x^{2}+x+1}{x+1}$$

Now, the next calculation will be:

$$x+\frac{x+1}{x^{2}+x+1}=\frac{x^{3}+x^{2}+2x+1}{x^{2}+x+1}$$

You then have:

$$x+\frac{x^{2}+x+1}{x^{3}+x^{2}+2x+1}=\frac{x^{4}+x^{3}+3x^{2}+2x+1}{x^{3}+x^{2}+2x+1}$$

Thus, we finally get:

$$1+\frac{x^{3}+x^{2}+2x+1}{x^{4}+x^{3}+3x^{2}+2x+1}=\frac{x^{4}+2x^{3}+4x^{2}+4x+2}{x^{4}+x^{3}+3x^{2}+2x+1}$$

 

[Krasz]

Thanks, but could you explain your reasoning for doing each step?

 

[arildno]

Not unless you point specifically at what you didn't understand.

 

[Krasz]

How did you get from step 2 to step 3? Such as how did 2x come into play?

 

[Borek]

How do you add fractions?

 

[arildno]

How did you get from step 2 to step 3? Such as how did 2x come into play?

After step 1, we have the following "bottom part":
$$x+\frac{1}{\frac{x^{2}+x+1}{x+1}}=x+\frac{x+1}{x^{2}+x+1}$$

Agreed thus far?

Now, we find a common denominator to the above sum:
$$x+\frac{x+1}{x^{2}+x+1}=\frac{x*(x^{2}+x+1)+x+1}{x^{2}+x+1}$$

Calculate the numerator of this expression!

 

[Krasz]

After step 1, we have the following "bottom part":
$$x+\frac{1}{\frac{x^{2}+x+1}{x+1}}=x+\frac{x+1}{x^{2}+x+1}$$

Agreed thus far?

Now, we find a common denominator to the above sum:
$$x+\frac{x+1}{x^{2}+x+1}=\frac{x*(x^{2}+x+1)+x+1}{x^{2}+x+1}$$

Calculate the numerator of this expression!

Thank you so much I see it now! You were a great help!

 

[arildno]

Even though they have not been assigned as exercises, I am sure your textbook contains a few more problems of the same type.

Do some of them to make sure you master the technique on your own!

Good luck!