Question

prove that the sequnce <an> defined by an=3n+7/4n+8 is a monotonic sequence.

Answer 1

Question #15204 Prove that the sequence $\{a_n\}_{n \geq 1}$ defined by $a_n = \frac{3n + 7}{4n + 8}$ is a monotonic sequence.

Solution. One has that for $n \geq 2$, $a_n - a_{n-1} = \frac{3n + 7}{4n + 8} - \frac{3n + 4}{4n + 4} = \frac{1}{4} \left( \frac{(3n + 7)(n + 1) - (3n + 4)(n + 2)(n + 1)}{(n + 2)(n + 1)} \right)$

$$\frac{3n^2 + 10n + 7 - 3n^2 - 10n - 8}{4(n + 2)(n + 1)} = -\frac{1}{4(n + 2)(n + 1)},$$

thus $\{a_n\}_{n \geq 1}$ is strictly decreasing sequence,