Increasing sequences

Write down the first four terms of the sequence $a_n = 3n - 5$ and check, by calculating the difference $a_{n+1} - a_n$, that it is a strictly increasing sequence.

Determine whether the sequence $c_n = n^2 + 2n$ is increasing. To do so, calculate algebraically the difference between term $n+1$ and term $n$, and analyse the sign of the result.

Prove algebraically that the sequence with general term $a_n = \frac{3n-1}{2n+5}$ is strictly increasing. To do so, expand the inequality $a_{n+1} - a_n > 0$ and justify the signs of the resulting numerator and denominator.

Given the geometric sequence $d_n = 3^n$, use the quotient method (that is, calculate $\frac{d_{n+1}}{d_n}$) to prove that it is a strictly increasing sequence.

The population of a bacteria colony in a controlled culture is modelled by the sequence $P_n = 200 \cdot (1,15)^n$, where $n \geq 1$ represents the number of days elapsed. Use the ratio test $\frac{P_{n+1}}{P_n}$ to prove rigorously that the population is constantly growing.

Study the monotonicity of the sequence $x_n = \frac{2^n}{n!}$. Show that it is not strictly increasing for all $n$, indicating from which specific term onwards it begins to strictly decrease.

Consider the sequence with radicals $c_n = \sqrt{n+5}$. Prove that it is strictly increasing. To avoid working directly with inequalities of surds, multiply and divide the expression $c_{n+1} - c_n$ by its conjugate.

Prove that the sequence with general term $a_n = \sqrt{n^2+n} - n$ is strictly increasing. Hint: Multiply and divide by the conjugate expression before analysing the difference $a_{n+1} - a_n$.

Find the values of the parameter $k \in \mathbb{R}$ for which the sequence defined by $a_n = (k-3)n + 5$ is strictly increasing. Reason your answer from the difference $a_{n+1} - a_n$.

Find the range of values of $x \in \mathbb{R}$ for which the geometric sequence $a_n = (2x - 1)^n$ is strictly increasing and has exclusively positive terms.