Mathematical Induction
- Thread starter Linthoi Y
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Steven G
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(k+1)4 =k4 +4k3 +6k2+4k +1
What can you say about each term in the prior line and 10k ?
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Hello, and welcome to FMH! 
Having presumably shown the base case is true, let's state the induction hypothesis \(P_k\):
[MATH]k^4<10^k[/MATH]
Now, for the induction step, I would look at:
[MATH](k+1)^4-k^4=4k^3+6k^2+4k+1[/MATH]
[MATH]10^{k+1}-10^k=9\cdot10^k[/MATH]
By our hypothesis, we must have that:
[MATH]9k^4<9\cdot10^k[/MATH]
So, can we show that:
[MATH]4k^3+6k^2+4k+1<9k^4[/MATH]
or equivalently:
[MATH](k+1)^4<10k^4[/MATH]
If we can demonstrate this to be true, then we can add to following to our induction hypothesis:
[MATH](k+1)^4-k^4<9\cdot10^k[/MATH]
Having presumably shown the base case is true, let's state the induction hypothesis \(P_k\):
[MATH]k^4<10^k[/MATH]
Now, for the induction step, I would look at:
[MATH](k+1)^4-k^4=4k^3+6k^2+4k+1[/MATH]
[MATH]10^{k+1}-10^k=9\cdot10^k[/MATH]
By our hypothesis, we must have that:
[MATH]9k^4<9\cdot10^k[/MATH]
So, can we show that:
[MATH]4k^3+6k^2+4k+1<9k^4[/MATH]
or equivalently:
[MATH](k+1)^4<10k^4[/MATH]
If we can demonstrate this to be true, then we can add to following to our induction hypothesis:
[MATH](k+1)^4-k^4<9\cdot10^k[/MATH]
Thank you so much. This is just what I neededHello, and welcome to FMH!
Having presumably shown the base case is true, let's state the induction hypothesis \(P_k\):
[MATH]k^4<10^k[/MATH]
Now, for the induction step, I would look at:
[MATH](k+1)^4-k^4=4k^3+6k^2+4k+1[/MATH]
[MATH]10^{k+1}-10^k=9\cdot10^k[/MATH]
By our hypothesis, we must have that:
[MATH]9k^4<9\cdot10^k[/MATH]
So, can we show that:
[MATH]4k^3+6k^2+4k+1<9k^4[/MATH]
or equivalently:
[MATH](k+1)^4<10k^4[/MATH]
If we can demonstrate this to be true, then we can add to following to our induction hypothesis:
[MATH](k+1)^4-k^4<9\cdot10^k[/MATH]