MATH135 Lecture 8: Lecture 8 Lecture 8, binomial theorem

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Mike Eden

Document Summary

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 12 + 22 + + n2. Last time we introduced the notation n! r! (n r)! and did a few calculations. If a and b are any numbers and n p, then (cid:18)n (cid:19) Alternatively, we can write (a + b)n = (cid:18) n n 1 (cid:19) (cid:18)n (cid:19) n bn abn 1 + (cid:18)n (cid:19) r an rbr + + an rbr. an 1b + + nx (cid:18)n (cid:19) r r=0. We will prove this and do some calculations, but need to do look at a couple of preliminary re- sults rst. If n and r are integers with 0 r n, then is an integer. (cid:18)n (cid:19) r.