Question

demonstrati ca sirul difinit prin formula
$an = \frac{n + 1}{n}$
este descrescator.
va rogggggg

Answer 1

$\text{Cel mai simplu ar fi sa aratam ca }\dfrac{a_{n+1}}{a_n}<1\\ \dfrac{a_{n+1}}{a_n}=\dfrac{\frac{n+2}{n+1}}{\frac{n+1}{n}}=\dfrac{n+2}{n+1}\cdot \dfrac{n}{n+1}=\dfrac{(n+2)n}{(n+1)^2}=\dfrac{n^2+2n}{n^2+2n+1}\\ \text{Mai departe avem ca:}\\ \dfrac{n^2+2n}{n^2+2n+1}<1\\ n^2+2n<n^2+2n+1\\ 0<1(A)\Rightarrow \text{sirul este descrescator}$