Can 4^n always be greater than or equal to n^4 for n ≥ 5?

[IntroAnalysis]

Homework Statement

n is an element of the Natural numbers, and n $\geq5$, then 4^n $\geq(n^4)$

Homework Equations

Base case: n=5, then 4^5=1024 >or= 5^4=625 as required.

The Attempt at a Solution

Inductive step, assume k is an element of Natural numbers and 4^k > or = k^4.
Then we must show 4^(k+1) > or = (K+1)^4

What's the trick to showing this?

 

[SammyS]

Homework Statement

n is an element of the Natural numbers, and n $\geq5$, then 4^n $\geq(n^4)$

Homework Equations

Base case: n=5, then 4^5=1024 ≥ 5^4=625 as required.

The Attempt at a Solution

Inductive step, assume k is an element of Natural numbers and 4^k ≥ k^4.
Then we must show 4^(k+1) ≥ (k+1)^4

What's the trick to showing this?
1. The problem statement, all variables and given known data

Homework Equations

The Attempt at a Solution

What have you tried?

 

[IntroAnalysis]

I know that 4^(k + 1) =4(4^k) which helps since we know 4^k >/ k^4,

But (k + 1)^4 = k^4 + 4k^3 + 6k^2 + 2k + 1, and I don't know how to break this up

to show 4(4^k) >/ k^4 + 4k^3 + 6k^2 + 2k + 1
Any hints?

 

[shaon0]

I know that 4^(k + 1) =4(4^k) which helps since we know 4^k >/ k^4,

But (k + 1)^4 = k^4 + 4k^3 + 6k^2 + 2k + 1, and I don't know how to break this up

to show 4(4^k) >/ k^4 + 4k^3 + 6k^2 + 2k + 1
Any hints?

RTP: 4ⁿ>n⁴ for all n≥5.
I'll skip the trivial n=5 case.

S(k): Assume; 4^k>k⁴ for all k≥5.

S(k+1): RTP: 4^k+1>(k+1)⁴
LHS=4^k+1=4^k*4>k⁴*4=4k⁴

Now consider; y=4x^4-(x+1)^4; Differentiate this and see what you can deduce for x≥5. Remember to restrict y after this to use on the induction. In restricting, change the domain and co-domain to an integer field.

 

[SammyS]

shaon0,

Do you realize that you are not supposed to give complete solutions?

 

[shaon0]

shaon0,

Do you realize that you are not supposed to give complete solutions?

Yeah, sorry. Uhm, just told by a mod. Deleting the post. Sorry. He's not online, so hopefully he hasn't seen it. I've deleted it and left a hint.

 

[IntroAnalysis]

Actually, I think I found an easier solution. Can someone comment?

We previously proved that for all n >/5, 2^n> n^2. We proved 2^(n+1) > (n+1)^2.
Now since we know that 2 > 0 and n + 1 > 0 (since n >/5), we know that we can square both sides and the inequality still holds so: (2^(n + 1))^2 = 2^(2n + 2) = 2^[2*(n+1)]
=4^(n+1) > ((n+1)^2)^2 = (n+1)^4
Hence, we have 4^(n+1) > (n+1)^4 which is what we wanted to prove.

 

[shaon0]

Actually, I think I found an easier solution. Can someone comment?

We previously proved that for all n >/5, 2^n> n^2. We proved 2^(n+1) > (n+1)^2.
Now since we know that 2 > 0 and n + 1 > 0 (since n >/5), we know that we can square both sides and the inequality still holds so: (2^(n + 1))^2 = 2^(2n + 2) = 2^[2*(n+1)]
=4^(n+1) > ((n+1)^2)^2 = (n+1)^4
Hence, we have 4^(n+1) > (n+1)^4 which is what we wanted to prove.

Yes, that's true but how would you have proven 2^n>n^2? As an extension; k^n>n^k for n,k_EZ⁺

 

[IntroAnalysis]

Base case n = 5: 2^n (= 32) > n^2 (=25) is true as required.

Now we assume that the statement is true for some n an element of Natural numbers, n$\geq5,$, and show that it must be true for n + 1. We know n²< 2ⁿ and therefore 2n²<2ⁿ⁺¹.
We are now going to use that 2n + 1 < n² which is true for all
n$\geq5$:

2n + 1 < n² implies n²+2n +1 < 2n²< 2ⁿ⁺¹ and therefore (n + 1)²< 2ⁿ⁺¹.

We are therefore done if we can demonstrate that 2n +1 < n² for all n $\geq5$. This inequality is equivalent to 2 < (n - 1)². The right side
is strictly increasing function of n for n > 1, thus (n - 1)²>(5 - 1)²= 16 > 2 for all n $\geq5$. This completes the proof of the original inequality.

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By the way, what does RTP stand for?

 

[IntroAnalysis]

Now consider; y=4x^4-(x+1)^4; Differentiate this and see what you can deduce for x≥5. Remember to restrict y after this to use on the induction. In restricting, change the domain and co-domain to an integer field.

So y = 4x⁴ - (x+1)⁴ = 4x⁴ - (x⁴ + 4x³ + 6x² + 4x +1)

Then dy/dx = 12x³ -12x² -12x -4 = 12x(x²-x -1) - 4
For x ≥ 5, (actually for x ≥ 2), dy/dx is positive. Thus, 4^(x+1)must be greater than (x + 1)⁴. Is that it?