binomial theorem

1. Prove that 13 + 2 + … +n= (1 + 2 + … + n) for all n € N. Hint: you may use any results from lectures to help out. If you do, please state them appropriately. 2. (a) Recall that, for n € N, n! :=nx(n-1) X… x 2 x 1 and given k € {0, 1, …, n}, we define n! (n – k)!K! Show that (%) + (-1) = (“!) for k € {0, 1,…, n}. (b) Using induction, prove the binomial theorem: for any r, y E R and n > 0, (7) (x+y)” = () ynt. k=0 3. Let F be an ordered field and for any z E F, define S. if r> 0 – if r 0. Show that | |
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