Your inequality is very similar to the Bernoulli's inequality.
I am having a hard time getting started on this.
n*3^{n}<= 4^{n}-2
where n>6
I have done the start step setting n = 7 and affirming that it is true.
and I assume
k*3^{k}<= 4^{k}-2
but I cannot grasp how I should prove
(k+1)*3^{(k+1)}<= 4^{(k+1)}-2
Thank you
Your inequality is very similar to the Bernoulli's inequality.