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October 1

Recurrence

Can someone please explain this to me: I have the Ramsey function $R(m,n)\leq R(m-1,n)+R(m,n-1)$ . I want to show that $R(m,n)\leq {\binom {m+n-2}{m-1}}$ . My book says that use a certain binomial coefficient. Can someone help? Thanks--Shahab (talk) 07:44, 1 October 2008 (UTC)[reply]

Oh never mind. Pascal's identity was what I was looking for. The proof was by double induction.--Shahab (talk) 08:11, 1 October 2008 (UTC)[reply]

algebra

They gave me this equation to use and I was ookay with it for a while. x = a₁ b₁ + a₂ b₂ / a₁ + a₂ but now they expect me to find b₂!

I know I should bring it around so that the one side is x/a₁ b₁ + a₂

But what do I do with a₁ + a₂? Do I put it so that x + (a₁ + a₂) or should I do it like x(a₁ + a₂)? --Jeevies (talk) 15:24, 1 October 2008 (UTC)[reply]

Step by step, apply the same operation to both sides. Adding parentheses and operators as my assumption of what you mean with your equation we have
x = ( ( ( a₁ * b₁ ) + ( a₂ * b₂ ) ) / a₁ ) + a₂.
Thus your first step would transform to the equation
( x - a₂ ) = ( ( ( a₁ * b₁ ) + ( a₂ * b₂ ) ) / a₁ ) + a₂ - a₂ = ( ( ( a₁ * b₁ ) + ( a₂ * b₂ ) ) / a₁ ).
Carry on from there. -- SGBailey (talk) 15:39, 1 October 2008 (UTC)[reply]

From the rest of the question I'm guessing they actually mean

x = (a₁ b₁ + a₂ b₂) / (a₁ + a₂)

That's the only way I can interpret the first steps they took. If I'm right the first step would be to multiply each side by (a₁ + a₂). -- Mad031683 (talk) 20:33, 2 October 2008 (UTC)[reply]