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.
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