Study Guide: Chapter 6 - Sequences and Series

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Chapter Six of the curriculum is dedicated to the study of "Sequences and Series" (المتتابعات والمتسلسلات). It provides a comprehensive exploration of mathematical patterns, their functional properties, and methods of formal proof.

Definition of a Sequence: A sequence is a function whose domain is the set of natural numbers ( $n \in \{1, 2, 3, \dots\}$ ) or a subset thereof.
The Variable $n$ : In this context, $n$ represents the position of the term in the sequence (the index).
The Term $a_n$ : This represents the value of the sequence at position $n$ , effectively the output of the function ( $f(n)$ ).
Functional Mapping: - Domain: Typically the set of positive integers ( $1, 2, 3, \dots$ ). - Range: The actual values (terms) of the sequence.
Types of Sequences Based on Domain: - Finite Sequence: A sequence that has a limited number of terms. - Infinite Sequence: A sequence that continues indefinitely without end.

Arithmetic Sequence Definition: A sequence in which the difference between any two consecutive terms is constant. This constant is known as the common difference ( $d$ ).
The General Term (Explicit Formula): - $a_n = a_1 + (n - 1)d$ - Where: - $a_n$ is the $n$ -th term. - $a_1$ is the first term. - $n$ is the number of the term. - $d$ is the common difference.
Arithmetic Series: The indicated sum of the terms of an arithmetic sequence.
Partial Sum of an Arithmetic Series ( $S_n$ ): - There are two primary formulas to calculate the sum of the first $n$ terms: - Formula 1: $S_n = \frac{n(a_1 + a_n)}{2}$ - Formula 2 (using $d$ ): $S_n = \frac{n[2a_1 + (n - 1)d]}{2}$

Geometric Sequence Definition: A sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero constant called the common ratio ( $r$ ).
The General Term (Explicit Formula): - $a_n = a_1 \cdot r^{n-1}$ - Where: - $a_n$ is the $n$ -th term. - $a_1$ is the first term. - $r$ is the common ratio (calculated as $\frac{a_n}{a_{n-1}}$ ). - $n$ is the term number.
Geometric Series: The indicated sum of the terms of a geometric sequence.
Partial Sum of a Geometric Series ( $S_n$ ): - The formula for the sum of the first $n$ terms of a geometric series is: - $S_n = \frac{a_1(1 - r^n)}{1 - r}$ - Condition: $r \neq 1$ .

Definition: A geometric series that continues without end ( $n \rightarrow \infty$ ).
Convergence and Divergence: - Convergent Series: If the absolute value of the common ratio is less than one ( $|r| < 1$ ), the series approaches a specific limit. - Divergent Series: If the absolute value of the common ratio is greater than or equal to one ( $|r| \geq 1$ ), the series does not have a finite sum.
Sum of an Infinite Geometric Series ( $S$ ): - For a convergent series, the sum is calculated using: - $S = \frac{a_1}{1 - r}$ - Condition: $|r| < 1$ .

Purpose: Provide a standardized formula for expanding binomial expressions raised to any non-negative integer power, such as $(a + b)^n$ .
The Binomial Theorem Formula: - $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ - Where: - $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$ .
Expansion Properties: - The number of terms in the expansion is $n + 1$ . - The sum of the exponents of $a$ and $b$ in each term is always equal to $n$ . - As you move through the terms, the exponent of $a$ decreases while the exponent of $b$ increases.
Pascal's Triangle: A triangular array of numbers that can be used to determine the coefficients of the binomial expansion.

Definition: A mathematical proof technique used to prove that a statement or formula holds for all natural numbers ( $n$ ).
The Principle of Mathematical Induction: - Step 1: The Basis Step: Show that the statement is true for the first natural number, typically $n = 1$ . - Step 2: The Inductive Hypothesis: Assume that the statement is true for an arbitrary natural number $k$ (Let $n = k$ ). - Step 3: The Inductive Step: Use the assumption from Step 2 to prove that the statement remains true for the next natural number, $n = k + 1$ .
Conclusion: If both the basis step and the inductive step are proven, then the statement is true for every natural number $n$ .