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 sequence with general term $b_n = \frac{n}{2} + 4$, prove that it is increasing by verifying that $b_{n+1} \geq b_n$ for any natural number $n$.

A student decides to save money for a trip. In the first month he saves $200$ pesos, and each subsequent month he saves $100$ pesos more than the previous month. Write down the general term for the money saved in month $n$ and justify mathematically why this sequence is strictly increasing.

Identify which of the following sequences is strictly increasing by calculating its first three terms and analysing its evolution: $x_n = -2n + 10$, $y_n = \frac{1}{n}$, $z_n = \frac{n+1}{n+2}$.

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.

A newly planted tree is $150$ cm tall. Each year, its height increases by $5$ % compared to the previous year's height. Express its height $h_n$ after $n$ years and justify using the ratio test that the sequence of its heights is increasing.

The population of a bacteria colony in a controlled culture is modelled by the sequence $P_n = 200 \times (1,15)^n$, 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.

In a kinematics experiment, the velocity of a moving object at time $n$ (in seconds) is given by the sequence $v_n = 10 - \frac{20}{n+1}$ in m/s. Prove analytically that the object is continuously accelerating (the sequence of velocities is strictly increasing) and observe the graph to determine whether it will exceed $10$ m/s at any point.

A technology company expects its monthly profits (in thousands of pesos) to follow the sequence $B_n = 1\,000 - \frac{2\,000}{2^n}$. Prove mathematically that profits increase from one month to the next.

In an industrial heating process, the temperature of a metal alloy at minute $n$ follows the law $T_n = 120 - 90(0,75)^n$ in degrees Celsius. Prove mathematically that the temperature is strictly increasing. From the graph, deduce whether the sequence is bounded above.

Prove that the sequence $b_n = \frac{n^2}{n^2+1}$ is strictly increasing for all $n \geq 1$. Is there any value of $n$ for which $b_n \geq 1$ holds? Justify your answer.

Prove that the rational sequence $a_n = \frac{pn+q}{rn+s}$ (assuming that $r > 0$ and $rn+s > 0$ for all $n \geq 1$) is strictly increasing if and only if the condition $ps - qr > 0$ holds.

Analyse the growth of the sequence $a_n = \frac{n^2+k}{n}$, where $k$ is a real number. Prove that the sequence is strictly increasing for all $n \geq 1$ if and only if the parameter $k$ satisfies the condition $k < 2$.

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.

Verify that the sequence defined by recurrence as $a_1 = 3$ and $a_{n+1} = a_n + \frac{1}{n}$ is strictly increasing for all $n \geq 1$.

Given the sequence defined as the partial sum $S_n = \sum_{i=1}^n \frac{1}{i^2} = 1 + \frac{1}{4} + \frac{1}{9} + \dots + \frac{1}{n^2}$, justify analytically why this sequence is strictly increasing and deduce whether the difference between consecutive terms increases or decreases as $n$ becomes very large.

A server network records the mean latency time (in milliseconds) using the recurrence sequence $u_1 = 2$, $u_{n+1} = \sqrt{6 + u_n}$. Given that all generated terms are positive and strictly less than $3$, prove that the sequence is strictly increasing by analysing the sign of $u_{n+1}^2 - u_n^2$.

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 $a_n = \left(1 + \frac{1}{n}\right)^n$. Given that this sequence models the growth of capital under 100% annual compound interest with $n$ compounding periods (whose limit is continuous compounding), calculate its first three terms to check empirically that it is increasing. Then, set up the start of its general proof by expanding the binomial theorem for $a_n$ and $a_{n+1}$.