Which of the following is bigger? $50^{50}$ or $49^{51}$

Question

Which of the following is bigger? $50^{50}$ or $49^{51}$

My attempt:

$\displaystyle\frac{49^{51}}{50^{50}}=\frac{{49}^{50}\cdot 49}{50^{50}}=\left(\frac{49}{50}\right)^{50}\cdot 49=(0.98)^{50}\cdot 49=(1-0.02)^{50}\cdot 49$

I used a calculator to show that $(1-0.02)^{50}\cdot 49>1$, but I don't know how to prove this algebraically. I know that $(1-x)^n\ge 1-nx$ for all natural numbers $n$ and all $x<1$, but that only shows that $(1-0.02)^{50}>0$, which is obvious.

Please help.

Answer 1

It's well known that $(1+\frac 1 n)^n<e$ for all $n$, in particular it's increasingly convergent to $e$, the Euler constant. So $(1+\frac 1 {49})^{50}<\frac{50e}{49}<3<49$, therefore $50^{50}<49^{51}$.