So I have been thinking about different ways/strategies of proving monotonicity of sequences, as I said in the title, I do not have a problem in mind but I want to hear different ways I may use during an exam. I will start off by listing a few:
$1)$ finding the consecutive differences between two terms and prove that it's always negative/positive.
$2)$ If a sequence is strictly positive/negative we can deduce the monotonicity by calculating the ratios between two consecutive terms and see if the ratio is less than or greater than $1$.
The followings are from the comments
$3)$Apply derivative test if the function is differentiable.
Given a sequence $\{a_n\}_{n\in\mathbb{N}}$ we can examine the associated function $a(x)$, take the derivative, and use the sign to determine if it's an increasing or decreasing function. This will of course imply the sequence itself is increasing or decreasing.
Simple example: $a_n = \frac{1}{n}$, associated function is $a(x) = \frac{1}{x}$. Taking the derivative on $]0,\infty[$ gives $a'(x) = -\frac{1}{x^2} < 0$ so $a_n$ is decreasing monotonically.