Help proving an inequality with induction

Question

I need to prove that 6n⁴> (n+2)⁴ for n ≥ 4

I've gotten this far,

First i prove that n=4 is true

When n=4, lhs=1536 and rhs =1296, so this works for n=4

now let k ≥ 4 so assume that P(k) is true so

6k⁴>(k+2)⁴

Now we need to show that p(k+1) is thrue that is,

6(k+1)⁴>(k+3)⁴

This is where i get stuck :-(

Please can someone show how i get to the end and show the inequality is true.

Thank you

Kara :-)

Answer 1

6(k+1)^4 - (k+3)^4

6(k^2+2k+1)(k^2+2k+1) - (k^2+6k+9)(k^2+6k+9) =

6( k^4 + 2k^3 + k^2

2k^3 + 4k^2 + 2k

k^2 + 2k + 1)

6 ( k^4 + 4k^3 + 6k^2 + 4k + 1) - (k^4 + 6k^3 + 9k^2

6k^3 + 36k^2 + 54k

9k^2 + 54k + 54)

6( k^4 + 4k^3 + 6k^2 + 4k + 1) - (k^4 + 12k^3 + 54k^2 + 108x + 54)

6k^4 + 24k^3 + 36k^2 + 24k + 6 - k^4 - 12k^3 - 54k^2 - 108k - 54

5k^4 +12k^3 -18k^2-84k - 48

this polynomial function has at most 1 positive solution since

there is only one sign change, per DeCartes rule of signs...

the graph crosses the x-axis at x=2.465, and stays positive

forever more...

so 6(k+1)^4 - (k+3)^4 > 0 for x>=4

therefore 6(l+1)^4 > (k+3)^4