Determine $n\in N$ such that $\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} \leq \frac{5^n-2^n}{5^n+2^n}$ .

Question

the question

Determine $n\in N$ such that $\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} \leq \frac{5^n-2^n}{5^n+2^n}$ .

The idea

So I thought of using these formulas;

$$a^n-b^n=(a-b)(a^{n-1}+...+b^{n-1})$$

and

$$a^n+b^n=(a+b)(a^{n-1}-...+b^{n-1})$$

The last one works only for even $n$ and it won't help me that much...

I Don't know where to start. I hope one of you can help me! Thank you!

Answer 1

$$\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} \leq \frac{5^n-2^n}{5^n+2^n}$$ $$\iff 1-\frac{2\cdot 2^n}{3^n+2^n}+1-\frac{2\cdot 3^n}{5^n+3^n} \leq 1- \frac{2\cdot 2^n}{5^n+2^n}$$ $$\iff \frac 12 \leq \frac{2^n}{3^n+2^n}+\frac{3^n}{5^n+3^n}-\frac{2^n}{5^n+2^n}$$ $$\iff \frac{3^n}{5^n+3^n} + 2^n\cdot\left(\frac1{3^n+2^n} - \frac1{5^n+2^n}\right) \ge \frac12$$ $$\iff \frac{3^n}{5^n+3^n} + \frac{10^n-6^n}{15^n+6^n+10^n+4^n} \ge \frac12$$Clearly, the left hand side is decreasing in $n$. However, when we set $n=0$, equality holds, so there are no solutions for $n \in \Bbb N$.

Answer 2

Are you sure that is the correct question?

if you look at this picture, I take $f(x)=\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} - \frac{5^n-2^n}{5^n+2^n}$ and I look for $f(x) \le 0 $ so that satisfy the $\frac{3^n-2^n}{3^n+2^n}+\frac{5^n-3^n}{5^n+3^n} \leq \frac{5^n-2^n}{5^n+2^n}$ but It seems there is no Natural solution for it!
You can use this link also https://www.desmos.com/calculator/gigrs3v682
Remark: For big $n \in \mathbb{N}$ there is no solution because $$\lim \frac{3^n-2^n}{3^n+2^n} \to 1\\ \lim \frac{5^n-2^n}{5^n+2^n} \to 1\\\lim \frac{5^n-3^n}{5^n+3^n} \to 1 \\ \to 1+1\le 1 \text{ no-solution}$$

Answer 3

For real numbers $a > b > c > 0$ is $$ \frac{a-b}{a+b} + \frac{b-c}{b+c} + \frac{c-a}{c+a} = \frac{(a-b)(b-c)(a-c)}{(a+b)(b+c)(a+c)} > 0 \, , $$ i.e. $$ \frac{a-b}{a+b} + \frac{b-c}{b+c} > \frac{a-c}{a+c} \, . $$ This holds in particular for $(a, b, c) = (5^n, 3^n, 2^n)$ with $n > 0$: $$ \frac{5^n-3^n}{5^n+3^n} + \frac{3^n-2^n}{3^n+2^n} > \frac{5^n-2^n}{5^n+2^n} $$ so the original inequality has no solutions with positive exponent $n$.

Answer 4

Consider the function $f(x)=\frac{x-1}{x+1}.$ For $x>3$, we have $\frac12<f(x)<1.$ Therefore, if $n\geq 3$, $$f((\frac32)^n)+f((\frac53)^n)>1>f((\frac52)^n) $$ and the given inequality is not satisfied. For $n=1,2$, it can be checked that it is not satisfied.