1. ## Induction of inequality

I am having a hard time getting started on this.

n*3n<= 4n-2
where n>6

I have done the start step setting n = 7 and affirming that it is true.

and I assume

k*3k<= 4k-2

but I cannot grasp how I should prove
(k+1)*3(k+1)<= 4(k+1)-2

Thank you

2. ## Re: Induction of inequality

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

3. ## Re: Induction of inequality

I fail to see how.

4. ## Re: Induction of inequality

Originally Posted by dowant
n*3n<= 4n-2 where n>6
I have done the start step setting n = 7 and affirming that it is true.
and I assume
k*3k<= 4k-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*}

5. ## 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*}