What is the key difference between an arithmetic and a geometric sequence definition?

An arithmetic sequence is defined by adding a constant difference ($u_{n+1} = u_n + r$), while a geometric sequence is defined by multiplying by a constant ratio ($u_{n+1} = q \cdot u_n$).

When is the method of comparing the ratio $\frac{u_{n+1}}{u_n}$ to 1 applicable for studying monotonicity?

This method is applicable when the terms of the sequence $u_n$ are strictly positive starting from a certain rank.

How can one find the general term $u_n$ if the sum $S_n$ is known?

The general term $u_n$ can be found by calculating $S_n - S_{n-1}$ for $n \ge 1$, provided $S_n$ is the sum from $u_0$ to $u_n$.

Numerical Sequences - Course and Corrected Exercises - 2Bac

Introduction to Numerical Sequences
📌 The course covers four main parts: numerical sequences (definition, boundedness, monotonicity, arithmetic/geometric), limits, convergence theorems, and recurrent sequences/adjacent sequences.
🔢 A numerical sequence $u_n$ is defined as an application from $\mathbb{N}$ or a subset of $\mathbb{N}$ to $\mathbb{R}$ , associating each integer $n \in \mathbb{N}$ with a real number $u_n$ .
📊 Examples provided include explicit sequences like $u_n = \sqrt{n}$ , recurrent sequences like $v_n = n^2$ , and simple sequences like $w_n = n$ .

Types of Sequences
💡 Explicit sequences are defined by their general term $u_n = f(n)$ , such as $u_n = 2n - 2$ .
🔄 Recurrent sequences are defined by their first term ( $u_{n_0}$ ) and a recurrence relation $u_{n+1} = f(u_n)$ .
🧩 Implicit sequences (for advanced math students) are solutions to an equation of the form $f_n(x) = \lambda$ , where $\lambda$ is a constant.

Boundedness of Numerical Sequences
🔗 A sequence $u_n$ (starting from $n_0$ ) is majorized if there exists a real number $M$ such that $u_n \le M$ for all $n \ge n_0$ .
📉 A sequence $u_n$ is minorized if there exists a real number $m$ such that $u_n \ge m$ for all $n \ge n_0$ .
⚖️ A sequence is bounded if it is both majorized and minorized.
🧪 Example: For $u_n = \frac{5n^2}{n^2 + 2}$ , the sequence is bounded by $m=0$ and $M=5$, since $u_n - 5 = \frac{-10}{n^2+2} < 0$ and $u_n \ge 0$ .

Monotonicity of Numerical Sequences
📈 A sequence $u_n$ is increasing if $u_{n+1} - u_n \ge 0$ (or $u_{n+1} \ge u_n$ ) starting from $n_0$ .
📉 A sequence $u_n$ is decreasing if $u_{n+1} - u_n \le 0$ (or $u_{n+1} \le u_n$ ) starting from $n_0$ .
🔁 Three methods to study monotonicity are presented: studying the sign of $u_{n+1} - u_n$ , comparing the ratio $\frac{u_{n+1}}{u_n}$ to 1 (if terms are positive), or using proof by induction.
🔗 If $u_n = f(n)$ , the monotonicity of $u_n$ follows the monotonicity of the function $f$ on the relevant interval.

Arithmetic and Geometric Sequences
➕ An arithmetic sequence is defined by $u_{n+1} = u_n + r$ , where $r$ is the constant common difference (reason).
✖️ The general term is $u_n = u_p + (n-p)r$ .
➗ A geometric sequence is defined by $u_{n+1} = q \cdot u_n$ , where $q$ is the constant common ratio (reason).
✖️ The general term is $u_n = u_p \cdot q^{n-p}$ .
🔗 Property for arithmetic: $2u_n = u_{n-1} + u_{n+1}$ .
🔗 Property for geometric: $u_n^2 = u_{n-1} \cdot u_{n+1}$ .

Application: Finding General Term from Sum
➕ Given the sum $S_n = \sum_{k=0}^{n} u_k = 4n^2 - 3n$ , the general term $u_n$ for $n \ge 1$ is found by $u_n = S_n - S_{n-1}$ .
🧮 Calculation yields $u_n = (4n^2 - 3n) - [4(n-1)^2 - 3(n-1)] = 8n - 7$ .
🎯 This derived sequence $u_n = 8n - 7$ is an arithmetic sequence with a common difference (reason) of $r=8$, and the first term ( $u_0$ ) is $-7$.

Key Points & Insights
➡️ To determine the general term $u_n$ from the sum $S_n$ for $n \ge 1$ , use the relation $u_n = S_n - S_{n-1}$ .
➡️ For sequences defined by $u_n = f(n)$ , study the monotonicity of the function $f$ to determine if the sequence is increasing or decreasing.
➡️ A strictly increasing sequence is always minorized by its first term ( $u_n \ge u_{n_0}$ ).
➡️ A strictly decreasing sequence is always majorized by its first term ( $u_n \le u_{n_0}$ ).

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Numerical Sequences - Course and Corrected Exercises - 2Bac – [Part 1]

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