Recent content by morbius27

Ok, so i worked on this a bit more, and found that the formula I'm trying to prove is Fn=F(n+2)-F(n+1), since this generates the negative terms...would the base case be n=1 and the induction hypothesis prove n=k-1?

Homework Statement Let the Fibonacci sequence Fn be defined by its recurrence relation (1) Fn=F(n-1)+F(n-2) for n>=3. Show that there is a unique way to extend the definition of Fn to integers n<=0 such that (1) holds for all integers n, and obtain an explicit formula for the terms Fn with...

well 30 of course =P. So what you're saying is that in order to cover the entire board, the T would have to cover two white and two black, but since it covers 1 white and 3 black (or vice versa) there will always be two left over from each color in the end?

Ahh, that does definitely make sense, but how can I show that the optimum way of putting down the Ts is by alternating centering them on black and white squares. i.e. why is that method favorable to, say, centering the first 8 on white then the rest on black, or something similar?

well, then obviously 15 Ts would be needed to cover the board, since there are 60 spots and 4 squares to a T. If you alternated centering them on white then black pieces, the 14th one would give 28 covered for both black and white. But since two of each are left, and a T requires 1 w/b and 3...

Homework Statement Consider an 8x8 checkerboard with two squares from each of two opposite corners deleted so that 60 squares are left (i.e the top row has 6 squares with the 2 far right squares missing, and the bottom row has 6 squares left with the 2 far left missing). Prove that the...

Well what I'm trying to ask is that given the statements in the initial post, what set of functions From R -->R satisfy them. The reason I ask about order is because i feel like the order in which the parts of the statements come affects the entire statement as a whole. The statements are...

But wouldn't the change in order of the epsilon and delta plus the switch between there exists and for all change the meaning of the statements?

Homework Statement Im trying to figure out what the difference is between the following two epsilon delta statements and the kinds of functions they satisfy: For all real numbers x and for all delta>0, there exists epsilon>0 such that |x|<delta implies |f(x)|<epsilon vs. there exists...

Thanks Tiny Tim! I got the solution after doing what you said and got the pattern; it worked beautifully after that ;)

Homework Statement Prove a simple formula for the number of closed intervals with integer endpoints contained in the interval [1,n] (including one point intervals), where n is a natural number. The Attempt at a Solution I know that the formula ends up being the sum of i from i=1 to i=n...

well I know you have to start with one side of the inequality by plugging in k+1 for n, and then manipulating it to the point where you can apply the induction hypothesis (2^n>=(n+1)^2) and then more manipulation to get to 2^(k+1)>=((k+1)+1)^2...I just don't know which side of the inequality to...

Hello, In this problem I am trying to Determine the exact set of natural numbers n for which the inequality 2^n>=(n+1)^2 holds. (equation (1)) I have already dealt with the base case where n=6, (since the inequality does not hold for n<6), and so (1) holds for n=k, and I need to show that it...