Question

Zbadaj monotoniczność ciągu
[tex]an=3-\frac{2}{3n+4}[/tex]

Answer 1

[tex]D: n\in N_+ \\ \\ a_n = 3 - \frac{2}{3n + 4} \\ \\ a_n + 1 = 3 - \frac{2}{3(n + 1) + 4} \\ \\ a_n + 1 - a_n = 3 - \frac{2}{3n + 7} - 3 - \frac{2}{3n + 4} \\ \\ a_n + 1 - a_n = \frac{9n + 21 - 2}{3n + 7} - \frac{9n + 12 - 2}{3n + 4} \\ \\ a_n + 1 - a_n = \frac{9n +19}{3n + 7} - \frac{9n + 10}{3n + 4} \\ \\ a_n + 1 - a_n = \frac{(3n + 4)(9n + 19) - (3n + 7)(9n + 10)}{(3n + 7)(3n + 4)} \\ \\ a_n + 1 - a_n = \frac{( {27n}^{2} + 57n + 36n + 76 ) - ( {27n}^{2} + 30n + 63n + 70 )}{(3n + 7)(3n + 4)} \\ \\ a_n + 1 - a_n = \frac{6}{(3n + 7)(3n + 4)} \\ \\ (3n + 7)(3n + 4) > 0 \leftrightarrow n\in N_+ \\ [/tex]

Ciąg jest rosnący.