Increasing Sequences

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

A student decides to save money for a trip. The first month they save $\$10$, and each successive month they save $\$5$ more than the previous month. Write the general term for the money saved in month $n$ and justify mathematically why this sequence is strictly increasing.

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

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

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

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 $ps - qr > 0$ is met.

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

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 $b_n \geq 1$ is satisfied? Justify your answer.

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 Fahrenheit. Prove mathematically that the temperature is strictly increasing. From the graph, deduce whether the sequence is bounded above.

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 measures $150$ inches. Each year, its height increases by $5$ % compared to the height of the previous year. Express its height $h_n$ after $n$ years and justify using the quotient criterion that the sequence of its heights is increasing.

The population of a colony of bacteria in a controlled culture is modeled by the sequence $P_n = 200 \cdot (1.15)^n$, where $n \geq 1$ represents the days elapsed. Use the quotient criterion $\frac{P_{n+1}}{P_n}$ to rigorously prove that the population is constantly increasing.

Consider the sequence with radicals $c_n = \sqrt{n+5}$. Prove that it is strictly increasing. To avoid working directly with inequalities of roots, 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 analyzing the difference $a_{n+1} - a_n$.

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 $u_1 = 2$, $u_{n+1} = \sqrt{6 + u_n}$. Knowing that all generated terms are positive and strictly less than $3$, prove by analyzing the sign of $u_{n+1}^2 - u_n^2$ that the sequence is strictly increasing.

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 based on 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.

Analyze 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 $k < 2$.

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

Let the sequence be $a_n = \left(1 + \frac{1}{n}\right)^n$. Knowing 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 empirically verify that it increases. Next, set up the beginning of its general proof by using the binomial expansion for $a_n$ and $a_{n+1}$.