Ok, that makes sense. It makes it a sum of 3 terms, the first two of which go to 0 and the last goes to 1. I think I tried that too, but with the first n-1 terms and that didn't work. I guess it never occurred to me to try it with the first n-2 terms. Thanks for your help.
Ok, here's my current solution. Like I said, it seems like an overly complicated way of doing it, and I wouldn't be surprised if there is some mistake in there somewhere.
Homework Statement
Find \stackrel{lim}{_{n\rightarrow\infty}}\frac{1^{1}+2^{2}+...+(n-1)^{n-1}+n^{n}}{n^{n}}.
2. The attempt at a solution
At first I split up the fraction into a sum of a bunch of terms, and said that all of the terms went to 0 except the last, which is 1. But then...
Homework Statement
This question was on a test in a math contest I was recently in, and I cannot seem to figure out how to get the answer:
Let f(x)=x7+x6+x5+x4+x3+x2+x+1. If k is a positive integer such that 53 < k < 115, find the sum of all distinct k such that the numerical remainder...
Homework Statement
If a stick is broken at random into 3 pieces, find the probability that the longest of the 3
pieces is at least three times the shortest of the 3 pieces and that the 3 pieces can serve as
the 3 sides of a triangle. Express your answer as a common fraction reduced to...
Homework Statement
A frictionless pulley connects 2 masses, one of which is on a frictionless inclined plane at angle θ, as shown in the diagram. The pulley system is set up so it has a mechanical advantage of 2 (so that if m1 moves 1 meter, m2 will move only 0.5 meters). Find equations...
Ah, that makes sense. So for each possible combination of 5 letters there are 5!=120 possible arrangements and only 1 that is in alphabetic order, so the probability is 1/120. It's interesting that it doesn't even matter how many tiles are in the bag, only how many you are picking at a time...
Homework Statement
There are 26 tiles in a bag, each with a different letter on it. You draw 5, without replacement. What is the probability that you pick the five letters in alphabetic order (e.g. C-H-R-T-W)
The Attempt at a Solution
The only way I could think of to do this was to find the...
Ok I think I got it now. Using y as the distance from the vertex to the center of the first circle, I got y/4 = (4+x)/x and y-8-x = 4+x. From the second equation I got y=2x+12, and plugging that into the first equation I found that x=4*sqrt(2). I guess I just didn't see that when I first tried...
Homework Statement
There are 3 circles, each tangent to 2 lines and to each other (as in the picture). The radius of the right (largest) circle is 8, and the radius of the left (smallest) circle is 4. What is the radius of the middle circle?
The Attempt at a Solution
I tried using...
Hi tiny-tim, thanks for your response! I also had that idea, but I could not figure out a general expression for that either. For k=5, the probability of not getting, say, card #1 by the fifth box is (0.8)^5. I thought I'd multiply that by 5 to cover the all 5 possible cards, but that adds to...
Homework Statement
A cereal box company started a promotion in which they put one of five different baseball cards in each box of cereal. All cards occur with equal probability. What is the expected number of cereal boxes you have to buy before you collect all five cards?
The Attempt at...