A Different Binomial Expansion TheoremDate: 07/23/98 at 09:22:44 From: Galip Adalan Subject: A new theorem about the famous binomial opening Dear Dr. Math: A few weeks ago, one of the students in a class that I teach asked me a brief and specific question about the famous binomial theorem. The student claims there is a completely different and and interesting form of the opening, which may be stated as following: (a+b)^n = (a^n + b^n) - T(n,1)*(a+b)^(n-2)*(a*b) + T(n,2)*(a+b)^(n-4)*(a*b)^2 - T(n,3)*(a+b)^(n-6)*(a*b)^4 + ... An alternating series, in which T(n,r) is a completely new coefficient of expansion. The student wants me to prove or state the theorem in the classroom. I haven't been able to locate the theorem. I'll owe my gratitude to you if you handle the case for me. Thanks in advance. Galip Adalan Date: 07/24/98 at 10:47:18 From: Doctor Rob Subject: Re: A new theorem about the famous binomial opening The correct formula is: (a+b)^n = a^n + b^n - Sum (-1)^r * T(n,r) * (a+b)^(n-2*r) * (a*b)^r where the sum runs over 1 <= r <= n/2. It is more usually given in the following variant form: a^n + b^n = Sum (-1)^r * T(n,r) * (a+b)^(n-2*r) * (a*b)^r where the sum runs over 0 <= r <= n/2, and n >= 1. The initial conditions and recursion that the T's satisfy is: T(n,0) = 1 for n >= 1 T(n,r) = 0 if n < 2*r T(n+1,r) = T(n,r) + T(n-1,r-1) This allows you to compute the values of T from the values with smaller n's and smaller or equal r's. First I would fix r = 0, and write down the values of T(n,0) = 1 in the first column. Then I would fill in 0's according to the second initial condition. Then I could compute T(n,1) for n = 2, 3, 4, .... Then I could compute T(n,2) for n = 4, 5, 6, ..., and so on. You can express the T(n,r) in terms of the binomial coefficients as follows: T(n,0) = C(n+1,0) T(n,1) = C(n,1) T(n,2) = C(n-1,2) - C(n-3,0) T(n,3) = C(n-2,3) - C(n-4,1) T(n,4) = C(n-3,4) - C(n-5,2) and so on. In general: T(n,r) = C(n-r+1,r) - C(n-r-1,r-2) You can verify that this expression for T(n,r) does satisfy the above recursion and initial conditions, and so must be equal to T(n,r). Using this, you can now prove the second formula given above using the Principle of Mathematical Induction, where you assume the statement true for n, then prove it for n+1. This proof is similar to the proof usually given for the Binomial Theorem. If this information is not sufficient to finish the proof, write back again and we can supply more detail. - Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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