Monotony of a sequence

Question

How do I prove that the function $n\to a_n:=\left(1+\frac{1}{n}\right)^{n+1}$ is decreasing, where $n$ is a positive integer?

P.s.: I'm 11th grade,and I didn't get to derivatives, we are currently studying limits of functions operations. Thanks in advance!

Answer 1

let $$a_n=\left(1+\frac{1}{n}\right)^{n+1}$$ and $$a_{n+1}=\left(1+\frac{1}{n+1}\right)^{n+2}$$ then calculate the Quotient $$\frac{a_{n+1}}{a_n}$$ see additionally here https://www.mathelounge.de/297952/nachweis-fur-monotonie-bei-folge-bn-1-1-n-n-1