Proving Induction: n4 <= 4n + 17 | Math Algebra Homework

[lovemake1]

Homework Statement

The induction question is. for all natural n, n⁴ <= 4ⁿ + 17Base case: 0 Works, since 0 < 1 + 17 then,
I assume that for all n in natural, n⁴ <= 4ⁿ + 17 holds.Now I believe I need to show that, 4(n⁴) <= 4(4ⁿ + 17)
that is, 4ⁿ⁺¹ + 17 >= (n+1)⁴
To do so, I prove, 4n⁴ >= (n+1)⁴,
which proves that 4ⁿ⁺¹ + 17 >= (n+1)⁴How would I prove.. 4n⁴ >= (n+1)⁴ = n⁴ + 4n³ + 6n² + 4n + 1

This step is in the middle of my induction proof and it is neccesary part of my induction step.

How would I go about doing this?
Some easier version similar to this deals with power of 2 or n, which seems rather simple. but, this one I am having hard time.
Help is much appreciated.

Homework Equations

The Attempt at a Solution

I tried starting from n⁴ = n⁴ and start adding things to both sides but the onlything I can add to left is n⁴ so I am not entirely sure how to go about doing this type of math. please some tricks and help is appreciated.

 

[SammyS]

Homework Statement

I am working on an induction problem but somehow I need to prove.
using algebra and such. 4n⁴ > (n+1)⁴ = n⁴ + 4n³ + 6n² + 4n + 1

This step is in the middle of my induction proof and it is necessary part of my induction step.

How would I go about doing this?

Homework Equations

The Attempt at a Solution

I tried starting from n⁴ = n⁴ and start adding things to both sides but the only thing I can add to left is n⁴ so I am not entirely sure how to go about doing this type of math. please some tricks and help is appreciated.

Please state the whole problem as it was given to you.

You're more likely to get help that way.

Also, at least sketch out your solution up to the point you're having trouble. It's hard for us to guess what you're trying to accomplish otherwise.

 

[skiller]

Agreed, for example this expression certainly isn't true for n=1, so for the induction to take place, where are you starting from?

 

[lovemake1]

Agreed, for example this expression certainly isn't true for n=1, so for the induction to take place, where are you starting from?

I added the full question with my understandings.

 

[lovemake1]

Agreed, for example this expression certainly isn't true for n=1, so for the induction to take place, where are you starting from?

sorry I fixed the problem. it's >= instead of >
thanks,

 

[skiller]

I added the full question with my understandings.

But you've changed the whole question since you last asked it. It is now a mess. Unless you yet again change it.

And n⁴⁺¹ is not the same as 4n⁴ which is how you have it written here, before you change it again!