3U inequality induction help (1 Viewer)

[abdullion]

Please help, c: - I don't know how to do it, so if someone could show the step by step working it'd be great.
10^n>=7^n+5^n for all integers n>=2

 

[glittergal96]

Let P(n)= 10^n - 7^n - 5^n.

Then
P(2)=100 - 49 - 25 > 0.

Now suppose P(n) > 0.

Then
P(n+1)=10^(n+1)-7^(n+1)-5^(n+1)
>10^(n+1)-10*7^n-10*5^n
=10*P(n)
>0.

Hence P(n) > 0 for all positive integers n >= 2 by induction.