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The answer to 8 is 500...
9. Suppose 0< a,b,c < 1 and ab + bc + ca = 1. Find the minimum value of a + b + c + abc.
I've tried using AM-GM on this and I can get an inequality for ab+bc+ca and a+b+c in respect to abc but I'm not sure how to answer the problem...
7. Let
and denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of .8. An ellipse and a hyperbola have the same foci, A and B, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let P be a point on both the hyperbola and ellipse. What is PA * PB?
@bobby: For number 4 I went back and I found that if you set the entire thing equal to some number n and put in exponent form you can solve for both x and y^5z with some manipulation. It can be done using trial and error, but there is a cleaner way.
@thickhead and ElainaVW:
I understand how to do number 5 now - I personally think it's more important to understand how to do it than to know the exact answer when you're learning, but in applied mathematics a small mistake can be fatal.
Yes, they were all correct. I understand how to get 3 and 5 now, but I still don't get 4. Thanks for all your help!
6. Find all ordered pairs of real numbers
such that .How did you get the answer to the above?
Let $$N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor). $$
Find $N$.
How do I use LaTeX?
Hi aleph_zero;
There is a unique ordered pair (c,d) such that c*phi^n + d^*phi hat ^ n$ is the closed form for sequence A_n.
What does ^* mean?
Sorry, that should just be a * (multiplication)
@thickhead
I can't find x though...
@bobby
I made a mistake simplifying one of the fractions in Problem 2...Thanks.
3.
Let
A_0 = 6
A_1 = 5
A_n = A_{n - 1} + A_{n - 2} for n ≥ 2
There is a unique ordered pair (c,d) such that c*phi^n + d^*phi hat ^ n$ is the closed form for sequence A_n.
Find c.
Note: phi is the golden ratio, phi hat is the complex conjugate
4. Let x, y, and z be positive real numbers that satisfy
2 log_x (2y) = 2 log_{2x} (4z) = log_{2x^4} (8yz) ≠ 0.
The value of xy^5z can be expressed in the form 1/(2^(p/q)), where p and q are relatively prime positive integers. Find p + q.
A sequence of real numbers (x_n) is defined recursively as follows: x_0=a and x_1=b are positive real numbers, and
x_(n+2) = (x_(n+1)+1)/(x_n)
for n = 0, 1, 2,.... Find the value of x_{2012}, in terms of a and b.
I've tried finding a pattern, but the calculations get messy pretty quickly. Help?
If a/b rounded to the nearest trillionth is 0.008012018027, where a and b are positive integers, what is the smallest possible value of a+b?
How do I do this problem? Thanks.
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