Answer
The sequence is eventually strictly increasing
Work Step by Step
Step 1. We're given the sequence \[ \{2n^2 - 7n\}_{n=1}^\infty \] and we're asked to show that the given sequence is strictly increasing or strictly decreasing.
Step 2. Let \[ f(x) = 2x^2 - 7x \] so that the $n$th term in the given sequence is \[ a_n = f(n). \]
Step 3. Let's find the first derivative, \[ f'(x). \] \[ f'(x) = 4x - 7 \] Hence, the function $f$ is increasing for $x > \frac{7}{4}$.
Step 4. Thus, \[ a_n = f(n) < f(n+1) = a_{n+1} \] for $n \geq 2$, which proves that the given sequence is eventually strictly increasing.
Result The given sequence is eventually strictly increasing.
