Re: Induction of inequality

Your inequality is very similar to the Bernoulli's inequality.

Re: Induction of inequality

Re: Induction of inequality

Quote:

Originally Posted by

**dowant** 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

Start from the above.

$\displaystyle \begin{align*}(k+1)3^{k+1}&=(3k+3)3^k \\&\le (3k+3)(4^k-2) \end{align*} $

Re: Induction of inequality

This can be proved using the auxiliary fact

$\displaystyle 3^{n+1}\le4^n$ for $\displaystyle n\ge4$ (*)

It is easily proved by induction.

$\displaystyle \begin{align*}(n+1)\cdot3^{n+1}&=3n\cdot 3^n+3^{n+1}\\ &\le3(4^n-2)+3^{n+1}&&\text{by IH}\\ &<3\cdot 4^n-2+3^{n+1}\\ &<3\cdot4^n+4^n-2&&\text{by (*))}\\ &=4^{n+1}-2 \end{align*}$