Question 338207: Use induction to show
1+2n_<3^n for n_>1
I'm really having a hard time understanding this process . I think if I see some examples it will help me solve it on my own
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 1+2n_<3^n for n_>1
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You show it is true for n = 1.
Then you assume it is true for n = k
and prove that is is true for n = k+1.
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If you can do that, you know it is true
for 1+1 = 2, 2+1=3, 3+1=4, etc.
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Your Problem
True for n = 1 ?
1+2*1 <= 3^1
1+2 <= 3
3<=3
True
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Assume true for n = k
1+2k <= 3^k
Note: We know it is true for k=1 because we did that above.
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Prove it is true for n = k+1
1+2(k+1) = (1+2k)+2
substitute 3^k for 1+2k to get
<= 3^k+2 <= 3^k+3 = 3^k*3 = 3^(k+1)
So 1+2(k+1) <= 3^(k+1)
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And that is what you were trying to prove
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So, 1+2n <= 3^n for all n>=1
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Cheers,
Stan H.
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