Mathematical Induction

Linthoi Y

New member
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

Elite Member
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 ?

• Linthoi Y

MarkFL

Super Moderator
Staff member
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

Linthoi Y

New member
(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

• MarkFL