Exercises: Decreasing sequence

Given the following sequences defined by their general term, identify which ones are strictly decreasing by empirically calculating their first four terms: $a_n = 5 - 2n$, $b_n = n^2 - 4$, $c_n = \frac{1}{2^n}$, $d_n = (-1)^n \times n$.

Prove, by analysing the sign of the difference $a_{n+1} - a_n$, that the polynomial sequence with general term $a_n = -4n + 7$ is strictly decreasing for every natural number $n$.

Check whether the rational sequence $b_n = \frac{3}{n+2}$ is decreasing. To do so, set up the expression $b_{n+1} - b_n$ and determine whether the result is less than or equal to zero for any $n \ge 1$.

Use the ratio test $\frac{a_{n+1}}{a_n}$ to prove that the geometric sequence of positive terms $a_n = \left(\frac{3}{4}\right)^n$ is strictly decreasing.

A sequence is defined recursively by the first term $a_1 = 15$ and the recurrence relation $a_{n+1} = a_n - \frac{1}{n}$. Justify mathematically why this sequence is strictly decreasing for all $n \ge 1$.

Calculate the first five terms of the quadratic sequence $c_n = -n^2 + 6n$. Is it a decreasing sequence from the first term? Determine from which value of $n$ onwards the sequence becomes strictly decreasing.

State whether the following statement is true or false and justify your answer with a numerical counterexample: "Every sequence whose terms are all negative is necessarily a decreasing sequence".

A mountaineer descends a mountain at a constant rate. Their altitude in metres at the end of each hour $n$ of the journey is modelled by the sequence $h_n = 3500 - 150n$. Prove analytically that the sequence of altitudes $(h_n)$ is strictly decreasing.

Cian states in Math class: «Since all the terms of the sequence $a_n = -\dfrac{4}{n+2}$ are negative numbers for any natural value $n \ge 1$, the sequence is necessarily strictly decreasing».

1. a) Identify Cian's conceptual error and provide a numerical counterexample by calculating its first three terms ($a_1$, $a_2$, and $a_3$) to refute his claim.
2. b) Formally prove the true monotonicity of the sequence by calculating the difference $a_{n+1} - a_n$.
3. c) Explain why Cian could not have used the ratio test $\dfrac{a_{n+1}}{a_n} < 1$ either to analyze this sequence.

To study the monotonicity of a sequence you have two main tools: studying the sign of the difference $a_{n+1} - a_n$ and the ratio test $\dfrac{a_{n+1}}{a_n} < 1$. Given the sequences $b_n = \dfrac{3n+8}{n+2}$ and $c_n = \dfrac{4^n}{n!}$:

1. a) Choose which of the two methods is more suitable for the sequence $b_n$, justify your choice, and prove that it is strictly decreasing for all $n \ge 1$.
2. b) Justify why for the sequence $c_n$ the ratio method is clearly more convenient than the difference method, check whether it satisfies the requirement regarding the sign of its terms in order to be applied, and prove that it is strictly decreasing starting from a certain term, indicating the first natural number $n$ from which it decreases.

Prove that the sequence with general term $a_n = \frac{2n+5}{3n+1}$ is strictly decreasing. To do so, set up the inequality $a_{n+1} < a_n$, perform algebraic operations and arrive at a statement that is true for all $n \ge 1$.

An industrial machine loses $12\%$ of its value each year relative to the immediately preceding year. If its initial purchase value is $45000$ euros, find the expression for the general term $V_n$ representing its value after $n$ years and prove mathematically that the sequence $(V_n)$ is decreasing.

Consider the sequence combining a power and a factorial $b_n = \frac{5^n}{n!}$. Calculate its first six terms and observe its behaviour. Using the ratio test, determine analytically from which exact value of $n$ the sequence becomes strictly decreasing.

Prove that the sequence constructed using radicals $c_n = \sqrt{n+2} - \sqrt{n}$ is strictly decreasing. Multiply and divide by the conjugate expression before analysing whether $c_{n+1} < c_n$.

A rubber ball is dropped from an initial height of $5$ metres. On each successive bounce, the ball reaches a height equal to $\frac{3}{4}$ of the maximum height of the previous bounce. Let $h_n$ be the maximum height reached after the $n$th bounce on the ground. Write the general term of the sequence and prove that it is strictly decreasing.

Given the polynomial sequence $a_n = -kn^2 + 8n - 3$, determine the set of values that the real parameter $k$ must take to ensure that the sequence is strictly decreasing for all $n \ge 1$.

The temperature in degrees Celsius of a piece of metal cooling in a room at room temperature is given by the sequence $T_n = 20 + 80 \times (0,85)^n$, where $n$ represents the time elapsed in minutes. Prove that $(T_n)$ is a decreasing sequence and calculate algebraically the expression for the exact temperature drop that occurs between minute $n$ and minute $n+1$.

Study the monotonicity of the sequence $d_n = \frac{(-1)^n}{n}$. Is it a decreasing sequence? Define a new sequence $e_n = |d_n|$ by taking the modulus of each term and prove step by step that $(e_n)$ is indeed a strictly decreasing sequence.

The Department of the Environment, Climate and Communications periodically monitors water reserves in reservoirs in Ireland. In a basin in the south of the country subjected to a dry summer period, the volume of stored water (in cubic hectometers, $\text{hm}^3$) over the weeks $n \ge 1$ elapsed since June 1 fits the model:

1. a) Formally prove, by means of the algebraic study of the difference $V_{n+1} - V_n$, that the water reserve strictly decreases week after week.
2. b) Determine whether the sequence is bounded below and bounded above. State the estimated initial maximum volume ($V_1$) and reason whether there exists a minimum water threshold below which the reservoir would never drop according to this model.
3. c) Calculate the amount of water lost between week 1 and week 2 ($V_1 - V_2$), and compare it with that lost between week 2 and week 3 ($V_2 - V_3$). Does the rate of weekly loss speed up or slow down as time passes?

Consider the sequence with general term $a_n = \dfrac{12n}{n^2 + 9}$ for $n \ge 1$.

1. a) Estimate: before performing formal algebraic calculations, compute the first four terms ($a_1, a_2, a_3, a_4$) and conjecture starting from which natural value of $n$ the sequence begins to be strictly decreasing, identifying which term seems to reach the maximum value.
2. b) Verify: formally set up the inequality $a_{n+1} - a_n < 0$, simplify the resulting fraction, and solve the inequality in the set of natural numbers to find with complete rigor the first term from which the sequence is strictly decreasing.
3. c) Assess whether your initial estimate coincided exactly with the algebraic result and explain in a reasoned manner why the degree of the denominator ends up dominating that of the numerator for sufficiently large values of $n$.

Find the rigorous mathematical condition that the real parameter $p$ must satisfy for the rational sequence $a_n = \frac{pn+4}{2n+3}$ to be strictly decreasing throughout its natural domain $n \ge 1$.

Let $(u_n)$ and $(v_n)$ be two sequences of strictly positive real numbers. Knowing that $(u_n)$ is strictly decreasing and $(v_n)$ is also strictly decreasing, prove algebraically that the product sequence defined by $w_n = u_n \times v_n$ is, in turn, strictly decreasing.

Consider the sequence of partial sums $S_n = \sum_{i=1}^n \frac{1}{3^i}$. Let $E_n = \frac{1}{2} - S_n$ be the sequence measuring the distance between $S_n$ and the theoretical limit of the infinite series. Prove that $(E_n)$ is a strictly decreasing sequence and express it in closed form in terms of $n$.

In a chemical reactor, the concentration $C_n$ of a radioactive isotope after $n$ hours is given by the recurrence relation $C_1 = 100$ and $C_{n+1} = 0.5 \times C_n + 10$. Prove by mathematical induction that $C_{n+1} < C_n$ for all $n \ge 1$, concluding that the sequence of concentrations is strictly decreasing.

The Babylonian method for approximating the value of $\sqrt{3}$ generates a sequence defined by $x_1 = 2$ and the recurrence relation $x_{n+1} = \frac{1}{2} \left( x_n + \frac{3}{x_n} \right)$. Given that all terms of the sequence satisfy $x_n > \sqrt{3}$, prove analytically that $x_{n+1} < x_n$ for all $n \ge 1$, thus showing that the sequence of approximations is strictly decreasing.

Study whether the logarithmic sequence $a_n = \ln\left(\frac{n+1}{n+2}\right)$ is increasing or decreasing. Then, prove that the sequence of partial sums $S_n = a_1 + a_2 + \dots + a_n$ is strictly decreasing and find a simplified algebraic expression for $S_n$ by applying the properties of logarithms.

Consider the sequence defined by $a_n = \frac{3^n + 2}{3^n}$. Prove that it is bounded below by $1$ and that it is strictly decreasing. Finally, determine from which value of $n$ onwards the difference between two consecutive terms has a modulus less than $10^{-4}$.

Consider the sequence defined by $a_n = \frac{n^n}{n! \times e^n}$. Given that the inequality $\left(1 + \frac{1}{n}\right)^n < e$ holds for any natural number $n$, prove using the ratio test that $(a_n)$ is a strictly decreasing sequence.

It is desired to design an algorithm that generates a sequence of numerical approximations with the structure $a_n = \dfrac{k \times n + 10}{n + p}$, where $k$ and $p$ are fixed real parameters. The algorithm must simultaneously satisfy three technical requirements:

1. The first generated value must be exactly $a_1 = 3$.

2. As the process advances ($n$ increasingly larger), the terms approach $2$ as closely as desired without ever becoming less than or equal to $2$ (that is, $2$ is the strict lower bound and limit of the sequence).

3. The sequence must be strictly decreasing for all $n \ge 1$.

1. a) Determine in a reasoned manner the unique values of the parameters $k$ and $p$ from requirements 1 and 2.
2. b) With the values found, rigorously prove using the definition of the difference $a_{n+1} - a_n < 0$ that the sequence satisfies requirement 3 for any $n \ge 1$.
3. c) Investigate what would happen to the monotonicity if, keeping the obtained value of $p$, the parameter $k$ were modified to become $k = 5$. Prove whether the new sequence would be strictly decreasing or strictly increasing.

Consider the sequence $a_n = \dfrac{2^n + 5}{2^{n+1} + 1}$ defined for all $n \ge 1$.

1. a) Rigorously prove that $(a_n)$ is strictly decreasing for all $n \ge 1$ by calculating the difference $a_{n+1} - a_n$. To operate clearly, keep the powers of base $2$ grouped together and justify the final sign of the resulting fraction.
2. b) Prove that the sequence is bounded above by its first term $a_1$ and bounded below by the value $m = 0.5$, algebraically proving that the difference $a_n - 0.5$ is strictly positive for any natural number $n$.
3. c) Supported by the decreasing nature and the found bound, determine starting from which natural number $n$ the distance between the term $a_n$ and the value $0.5$ becomes less than one hundredth ($0.01$).