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 meters at the end of each hour $n$ of the journey is modeled by the sequence $h_n = 3500 - 150n$. Prove analytically that the sequence of altitudes $(h_n)$ is strictly decreasing.

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 $900\,000$ pesos, 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 behavior. 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 $5$ meters. 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.

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.