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Thank you for the help.

I need to prove that for any full binary tree with N\geq1 leaves, that it has 2N-1 nodes. I can see this when looking at a tree, but I can't figure out how to prove this.

Oh bah, I messed up the formula. It should be correct now.

I know how to do an inductive proof, but I am just not sure how to prove this. I've never done an inductive proof on a piecewise function where the proof depends on whether the nth term is even or odd.

I have the series 2, 2, 6, 10, 22, 42, 86, 170, 342, 682 which follows the following: M(1)=2 M(2)=2 M(3)=M(1)+M(2)+2 M(4)=M(3)+M(2)+M(1) M(5)=M(4)+M(3)+M(2)+M(1)+2 M(6)=M(5)+M(4)+M(3)+M(2)+M(1) Each value for M(N) for N >= 3 is the sum of all previous values, and add 2 if N is odd...

Well, this August I will be an incoming freshman at college. I placed out of calculus 1 and 2, and will be starting with multivariable calculus, and then differential equations. I am well aware that professors do not allow the use of calculators on exams, but would still like to have one for...

Even if it were theoretically possible, no human or materials used in the spacecraft could survive such rapid acceleration and deceleration. Plus, it isn't cost effective to do. The few labs that can create antimatter do it at extreme costs, and very little quantity is produced that can be...

Well, since you know that the Taylor series for \sin x=x-\frac{x^3}{3!} + \frac{x^5}{5!}-\frac{x^7}{7!}+\cdots then you can just plug in x^2 for x in the Taylor expansion, so it would become:\sin x^2=x^2-\frac{x^6}{3!}. Now you can integrate f'(x) as the taylor approximation, with: \int...