Exercises: Increasing sequence

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.

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 $10$ euros, and each subsequent month he saves $5$ euros 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.

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.

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 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.

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

Lucía wants to prove that the sequence $a_n = 6n - n^2$ is strictly increasing. To do so, she calculates the first two terms: $a_1 = 6\cdot 1 - 1^2 = 5$ and $a_2 = 6\cdot 2 - 2^2 = 8$. Since $a_2 > a_1$, she firmly states that the sequence is strictly increasing for all $n \in \mathbb{N}$.

1. a) Explain precisely the conceptual error in reasoning that Lucía makes when deducing the global behavior from only two particular terms.
2. b) Find a counterexample by calculating the terms $a_3$, $a_4$, and $a_5$ to show that her general statement is false.
3. c) Write the formal algebraic definition that Lucía should have checked for all $n \geq 1$ and calculate the difference $a_{n+1} - a_n$ in terms of $n$ to determine from which value of $n$ the sequence stops increasing.

Consider the rational sequence $a_n = \dfrac{3n - 2}{n + 1}$ for $n \geq 1$.

1. a) Before carrying out any complex algebraic calculation, intuitively estimate whether the sequence will be increasing or decreasing by mentally calculating the approximate values of $a_1$, $a_2$, and a distant term such as $a_{100}$. Toward what number does the sequence seem to approach as $n$ becomes very large?
2. b) Formally prove that the sequence is strictly increasing by expanding the difference $a_{n+1} - a_n$, putting it over a common denominator, and justifying the sign of both the resulting numerator and denominator for any $n \geq 1$.
3. c) Check whether your initial estimate was correct and reason whether the distance between two consecutive terms, $a_{n+1} - a_n$, increases or decreases as $n$ grows.

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.

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.

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 $a_{n+1} - a_n$.

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.

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.

A technology company expects its monthly profits (in thousands of euros) to follow the sequence $B_n = 50 - \frac{100}{2^n}$. Prove mathematically that profits increase from one month to the next.

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.

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.

To prove that the sequence $a_n = \dfrac{2^n}{n + 1}$ is strictly increasing for all $n \geq 1$, two different algebraic paths are proposed:

• Method 1: Study the sign of the difference $a_{n+1} - a_n$.

• Method 2: Apply the ratio test by analyzing whether $\dfrac{a_{n+1}}{a_n} > 1$.

1. a) Explain why it is essential to verify that all terms of the sequence are strictly positive before being able to validly apply Method 2.
2. b) Fully carry out the proof following both methods, simplifying as much as possible the algebraic expressions obtained in each of them.
3. c) Compare both procedures and justify which of the two is more agile and practical when exponential expressions such as $2^n$ are involved.

Red Eléctrica de España (REE) monitors the incorporation of solar photovoltaic power into the mainland grid. A forecasting model for the cumulative installed power in an autonomous community over the coming semesters ($n \geq 1$) is given, in gigawatts ($\text{GW}$), by the general term $P_n = \dfrac{12n + 5}{2n + 9}$.

1. a) Analytically prove that the cumulative power will be strictly increasing throughout all semesters by showing that $P_{n+1} - P_n > 0$ for any $n \geq 1$.
2. b) A technical report claims that, since the power always grows from one semester to the next, over the years it will certainly exceed $7\text{ GW}$. Use your knowledge about the behavior of this sequence to confirm or refute this statement with reasoning.
3. c) Calculate the increase in power between the first and the second semester ($P_2 - P_1$) and compare it with the increase between the tenth and the eleventh ($P_{11} - P_{10}$). Interpret what it means in practice that the sequence grows but the difference between consecutive terms becomes smaller and smaller.

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 $k < 2$.

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.

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.

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 $a_{n+1} - a_n$.

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.

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 $u_{n+1}^2 - u_n^2$.

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 $a_{n+1}$.

Design a rational sequence of the form $a_n = \dfrac{a\cdot n + b}{n + 3}$ (where $a$ and $b$ are fixed integers) that simultaneously satisfies the following three conditions:

1. Its first term is negative: $a_1 = -1$.

2. It is strictly increasing for all $n \geq 1$.

3. No term of the sequence reaches or exceeds the value $4$ (that is, $a_n < 4$ for all $n \geq 1$).

To solve it:

1. a) Translate each of the three conditions into algebraic relations or inequalities involving the parameters $a$ and $b$.
2. b) Deduce the set of permissible integer values for $a$ and choose a specific pair $(a, b)$ that satisfies all the imposed conditions, writing the final general term of your sequence.
3. c) Prove with formal rigor, from the algebraic study of the difference $a_{n+1} - a_n$, that the sequence you have constructed is indeed strictly increasing for all $n \geq 1$.

Carlos attempts to prove that the sequence with radicals $a_n = \sqrt{n^2 + 5n} - n$ is strictly increasing for all $n \geq 1$. To study the difference $a_{n+1} - a_n$, he sets up:

$a_{n+1} - a_n = \left(\sqrt{(n+1)^2 + 5(n+1)} - (n+1)\right) - \left(\sqrt{n^2 + 5n} - n\right) = \sqrt{n^2 + 7n + 6} - \sqrt{n^2 + 5n} - 1$.

Not knowing how to justify the sign of this expression, he decides to square each term separately and writes:

$(\sqrt{n^2 + 7n + 6})^2 - (\sqrt{n^2 + 5n})^2 - 1^2 = (n^2 + 7n + 6) - (n^2 + 5n) - 1 = 2n + 5$.

Since $2n + 5 > 0$ for all $n \geq 1$, he concludes that the sequence is strictly increasing.

1. a) Explain in detail the algebraic fallacy committed by Carlos and show by means of an elementary numerical example why the operation he performed does not preserve the sign or the order of a subtraction of terms.
2. b) Correct the exercise by rigorously proving that the sequence is strictly increasing. To do this, rewrite the general term $a_n$ by multiplying and dividing by its conjugate expression, and deduce from said fraction that $a_{n+1} > a_n$ for all $n \geq 1$.
3. c) Reason analytically whether an upper bound exists for this sequence and find the smallest integer $M$ that satisfies $a_n < M$ for all $n \geq 1$.