Sequences/Series Sigma Question

[JBD2]

Homework Statement

If $$\sum_{n=1}^{4}log_{2}x^{n}=80$$, determine the value of x.

Homework Equations

$$S_{n} = \frac{a(r^{n}-1)}{r-1}$$

$$S_{n} = \frac{rt_{n}-a}{r-1}$$

The Attempt at a Solution

$$log_{2}1+log_{2}x^{2}+log_{2}x^{6}+log_{2}x^{12}=80$$?

$$log_{2}x^{20} = 80$$

$$2^{80}=x^{20}$$

$$x=16$$

Is this right? Or does x have to be + or - 16 or am I just completely wrong?

 

[Avodyne]

This looks right, assuming that $$log_{2}x^{n}$$ means $$\log_{2}(x^{n})$$ and not $$(\log_{2}x)^n$$.

 

[JBD2]

Ok well I have two more questions, and rather than clutter the board with another post I'll put them here:

Question 2:

Homework Statement

A new car costs $42 000 and depreciates 20% the first year, then 15% every year after. What is the car worth in 10 years?

The Attempt at a Solution

42 000 + 42 000(0.8) + 33600(0.85) + 33600(0.85)²...

I'm not sure if this is right so far, but I'm pretty sure I can separate the 42 000(0.8) and the 33600(0.85) right? What should I do?Question 3:

Homework Statement

In a geometric sequence, t₂+t₃=60 while t₄+t₅=1500. Find the first 3 terms.

The Attempt at a Solution

Not really sure how to start this one, I think that t4 is equal to 60r and t5 is equal to 60r², so can I go:

60r + 60r²=1500 and solve for r?

If I do this I end up getting $$\frac{-1\pm\sqrt{101}}{2}$$ which I'm sure can't be correct.

 

[Chaos2009]

Well, I don't think your first problem is a series. I think you are making it too hard on yourself. It starts at $42,000, loses 20%, and becomes $33600 after one year. I don't think you just keep adding them up.

 

[Avodyne]

I don't think you just keep adding them up.

If you're supposed to add them up, then I want that car!

As for #3, for a geometric series, $$t_n = a\,r^n$$ for some constants a and r. So you have two equations in two unknowns.