Mathematical Induction

Linthoi Y

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Jul 13, 2019
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I am having trouble solving this question that is n^4<10^n.I have the problem in solving for k+1 that is in the proof of (k+1)^4<10^(k+1). Please help me
 

Jomo

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Dec 30, 2014
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I am having trouble solving this question that is n^4<10^n.I have the problem in solving for k+1 that is in the proof of (k+1)^4<10^(k+1). Please help me
(k+1)4 =k4 +4k3 +6k2+4k +1
What can you say about each term in the prior line and 10k ?
 

MarkFL

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Hello, and welcome to FMH! :)

Having presumably shown the base case is true, let's state the induction hypothesis \(P_k\):

\(\displaystyle k^4<10^k\)

Now, for the induction step, I would look at:

\(\displaystyle (k+1)^4-k^4=4k^3+6k^2+4k+1\)

\(\displaystyle 10^{k+1}-10^k=9\cdot10^k\)

By our hypothesis, we must have that:

\(\displaystyle 9k^4<9\cdot10^k\)

So, can we show that:

\(\displaystyle 4k^3+6k^2+4k+1<9k^4\)

or equivalently:

\(\displaystyle (k+1)^4<10k^4\)

If we can demonstrate this to be true, then we can add to following to our induction hypothesis:

\(\displaystyle (k+1)^4-k^4<9\cdot10^k\)
 

Linthoi Y

New member
Joined
Jul 13, 2019
Messages
2
(k+1)4 =k4 +4k3 +6k2+4k +1
What can you say about each term in the prior line and 10k ?
Hello, and welcome to FMH! :)

Having presumably shown the base case is true, let's state the induction hypothesis \(P_k\):

\(\displaystyle k^4<10^k\)

Now, for the induction step, I would look at:

\(\displaystyle (k+1)^4-k^4=4k^3+6k^2+4k+1\)

\(\displaystyle 10^{k+1}-10^k=9\cdot10^k\)

By our hypothesis, we must have that:

\(\displaystyle 9k^4<9\cdot10^k\)

So, can we show that:

\(\displaystyle 4k^3+6k^2+4k+1<9k^4\)

or equivalently:

\(\displaystyle (k+1)^4<10k^4\)

If we can demonstrate this to be true, then we can add to following to our induction hypothesis:

\(\displaystyle (k+1)^4-k^4<9\cdot10^k\)
Thank you so much. This is just what I needed
 
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