SOLUTION: Prove by induction that 3^n ≥ 2n +1 for all positive integers.

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Question 909315: Prove by induction that 3^n ≥ 2n +1 for all positive integers.
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Prove 3%5En%3E=2n%2B1.

n=1 case: 3%5E1%3E=2%2A1%2B1
3%3E=3, yes.

ASSUME highlight_green%283%5Ek%3E=2k%2B1%29 to be true. k is any natural number.

PROVE that k+1 case is true:
3%5E%28k%2B1%29%3E=2%28k%2B1%29%2B1
3%5E%28k%2B1%29%3E=2k%2B2%2B1
3%5E%28k%2B1%29%3E=2k%2B3
3%2A3%5Ek%3E=2k%2B3
Divide both sides by 3
highlight_green%283%5Ek%3E=2k%2F3%2B1%29

Compare this last found inequality with the ASSUMPTION statement inequality.
%282%2F3%29k%3C2k as you can compare in the corresponding positions in each statement.
The latter highlighted-in-green statement is also then true if the assumption
statement is true.
PROVED.