Decreasing sequences

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 $n$.

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".

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 $n \ge 1$.

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.

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.

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.

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.

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 $c_{n+1} < c_n$.

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 $10^{-4}$.