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Unformatted text preview: MATH 135 Fall 2008 Lecture VIII Notes Binomial Theorem Recall that we are trying to come up with a way of expanding ( a + b ) n without actually having to expand it for each value of n in which we are interested. This is similar to wanting to come up with “closed form” expressions for things like 1 2 + 2 2 + ··· + n 2 . Last time we introduced the notation n r = n ! r !( n- r )! and did a few calculations. Binomial Theorem (Theorem 4.34) If a and b are any numbers and n ∈ P , then ( a + b ) n = n a n + n 1 a n- 1 b + ··· + n r a n- r b r + ··· + n n- 1 ab n- 1 + n n b n Alternatively, we can write ( a + b ) n = n X r =0 n r a n- r b r . We will prove this and do some calculations, but need to do look at a couple of preliminary re- sults first. Proposition 4.33 If n and r are integers with 0 ≤ r ≤ n , then n r is an integer. Rationale We will not formally prove this. However, last time we looked at n r as the number of ways of choosing r objects from among n objects. Since this number of ways is an integer, then n r should be an integer.be an integer....
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