Unit 1: Sequences and Series

This unit explores the fundamental properties and patterns of numbers arranged in specific orders, known as sequences, and their summations, called series. It establishes the mathematical framework for identifying arithmetic and geometric progressions, calculating partial and infinite sums, and analyzing convergence and divergence. These tools are applied to real-world scenarios, including population growth, financial interest, and physical measurements.

1.1 Sequences

• A sequence is an arrangement of numbers in a definite order according to some rule.
• A finite sequence is a sequence that has a last term; its domain is $\{1, 2, 3, \dots, n\}$.
• An infinite sequence is a sequence that does not have a last term; its domain is the set of natural numbers ($\mathbb{N}$).
• A recursive sequence is one where the general term $a_n$ relates to one or more terms that come before it.
• The Fibonacci sequence is a specific recursive sequence defined as $F_0=1, F_1=1,$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 2$.
• The Mulatu sequence is a recursive sequence defined as $M_0=4, M_1=1,$ and $M_n = M_{n-1} + M_{n-2}$ for $n \ge 2$.

1.2 Arithmetic and Geometric Sequences

• An arithmetic sequence (or arithmetic progression) is a sequence in which each term except the first is obtained by adding a fixed number to the preceding term.
• The common difference (d) is the fixed number added in an arithmetic sequence.
• The general term of an arithmetic sequence is given by $A_n = A_1 + (n – 1)d$, where $A_1$ is the first term.
• A geometric sequence (or geometric progression) is one in which the ratio between consecutive terms is a non-zero constant.
• The common ratio (r) is the constant ratio in a geometric sequence, calculated as $r = \frac{G_{n+1}}{G_n}$.
• The general term of a geometric sequence is given by $G_n = G_1 r^{n-1}$.
• The geometric mean ($m$) between two non-zero numbers a and b is $m = \pm \sqrt{ab}$.

Process: Inserting Arithmetic Means Between Two Numbers

1. Let the two given numbers be $A_1$ and $A_n$.
2. Determine the total number of terms (n) by adding the number of means to be inserted to the 2 given endpoints.
3. Use the formula $A_n = A_1 + (n – 1)d$ and substitute the values for $A_n, A_1,$ and n.
4. Solve for the common difference d.
5. Add $d$ successively to $A_1$ to find each of the inserted terms.

1.3 The Sigma Notation and Partial Sums

• A partial sum ($S_n$) is the sum of the first n terms of a sequence.
• Sigma notation (or summation notation) uses the Greek letter $\Sigma$ to write out a long sum concisely.
• A telescoping sequence is a sequence where partial sums simplify significantly because terms in the middle cancel out.
• The sum of the first n natural numbers is $S_n = \frac{n}{2}(n+1)$.
• The sum of the first n terms of an arithmetic sequence is $S_n = \frac{n}{2}[2A_1 + (n – 1)d]$ or $S_n = n(\frac{A_1 + A_n}{2})$.
• The sum of the first n terms of a geometric sequence (where $r \neq 1$) is $S_n = G_1 \frac{(1-r^n)}{1-r}$.

1.4 Infinite Series

• An infinite series is the sum of the terms of an infinite sequence, denoted $\sum_{n=1}^\infty a_n$.
• A convergent series is an infinite series where the sequence of partial sums $S_n$ approaches a real number S as n approaches infinity.
• A divergent series is an infinite series that does not converge to a finite real number.
• For a geometric sequence, the series converges if $|r| [cite_start]< 1$ and the sum to infinity is $S_\infty = \frac{G_1}{1-r}$.
• Recurring decimals can be written as the sum of an infinite converging geometric series and converted to fractions.

1.5 Applications of Sequence and Series in Daily Life

• Arithmetic sequences model situations with constant growth, such as fixed salary raises or fixed saving increases.
• Geometric sequences model rapid changes, such as compound interest or the half-life decay of radioactive substances.
• Summation formulas are used to calculate total earnings over time or determine if a physical growth (like a flower) will ever exceed a specific limit.

Key Terminology

• Arithmetic sequence: A sequence with a constant difference between consecutive terms.
• Common difference: The fixed number added to find the next term in an arithmetic progression.
• Common ratio: The non-zero constant ratio between terms in a geometric progression.
• Convergent series: An infinite series whose partial sums approach a finite limit.
• Divergent series: An infinite series that does not approach a finite limit.
• Finite sequence: A sequence that contains a specific last term.
• General term: The formula ($a_n$) used to find any term in a sequence based on its position.
• Geometric sequence: A sequence where consecutive terms have a constant ratio.
• Infinite sequence: A sequence that continues without a last term.
• Infinite series: The sum of terms from an infinite sequence.
• Natural numbers: The set $\{1, 2, 3, \dots\}$ used as the domain for infinite sequences.
• Partial sums: The sum of a finite number of terms from the beginning of a sequence.
• Recursion formula: A formula relating a term to those preceding it.
• Sequence: An arrangement of numbers in a specific, rule-based order.
• Series: The sum of the terms of a sequence.
• Terms of a sequence: The individual numbers that make up a sequence.