Use mathematical induction to prove that the statement is true for every positive integer n.(1 - ) (1 - ) . . . (1 - ) = 

What will be an ideal response?


Answers will vary. One possible proof follows.
a). Let n = 1. Then, (1 - ) =  =  = . Thus, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 Sk = .
 Also, if the statement is true for n = k + 1, then
 Sk+1 = Sk ? (1 - ) = 
 Substitute to get:
 ? (1 - ) = ,
 or
 ? (1 - ) = .
 Simplify:
 ? ( - ) = 
 ? () = 
  = 
Since the equality holds, then the statement is true for n = k + 1 as long as it is true for n = k. Furthermore, the statement is true for n = 1. Therefore, the statement is true for all natural numbers n.

Mathematics

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