Math 11: Unit 1 Patterns, Sequences, and Series - Definitive Study Guide

Unit 1: Patterns and Natural Resources

• Mathematical Applications in Real-World Contexts

  • Economic and Resource Trends: Often based on sequences and series to predict growth or decline.
  • Seismic Exploration: Identifies underground phenomena (caves, oil pockets, rock layers) by transmitting sound and timing vibration echoes.
  • Surveying: Uses triangulation and trigonometry laws to determine distances between inaccessible points.

• Unit 1 Project: Canada’s Natural Resources

  • Scope: Exploration of petroleum, minerals, and forestry.
  • Mathematics Application:
    • Chapter 1: Applying sequences and series to production data and development predictions.
    • Chapter 2: Using sine and cosine laws to explore resource discovery areas and proposed sites.
  • Resource Categories:
    • Petroleum: Includes the Athabasca Oil Sands (estimated $1.6 \times 10^{12}$ barrels).
    • Minerals: Includes gold, silver, potash, and diamonds. Canada is the world's largest potash producer.
    • Forestry: Canada possess $402.1 \times 10^6\,ha$ of forest, representing $41.1\%$ of its total surface area.

Chapter 1: Sequences and Series

• Patterns in Nature

  • Logistic Spirals: Patterns like the Golden Mean spiral, based on the Fibonacci sequence.
  • Found in: Chambered nautilus shells, inner ear structure, star clusters, cloud patterns, whirlpools, seed growth, leaf arrangements, and rabbit reproduction.
  • Fibonacci Sequence Definition: A sequence of natural numbers where each number is the sum of the two preceding numbers.
    • Verbatim sequence: $1, 1, 2, 3, 5, 8, 13, \dots$
  • Key Terms:
    • Sequence: An ordered list of elements.
    • Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
    • Common Difference ($d$): The constant value added to each term; $d = t_n - t_{n-1}$.
    • General Term ($t_n$): An expression for directly determining any term in a sequence.
    • Arithmetic Series: The sum of terms in an arithmetic sequence.
    • Geometric Sequence: A sequence where the ratio of consecutive terms is constant.
    • Common Ratio ($r$): The constant multiplier between terms; $r = \frac{t_n}{t_{n-1}}$.
    • Geometric Series: The sum of terms in a geometric sequence.
    • Convergent Series: An infinite series where the sequence of partial sums approaches a fixed value.
    • Divergent Series: An infinite series where the sequence of partial sums does not approach a fixed value.

• Career Link: Biomedical Engineering

  • Role: Combines biology, engineering, and math to solve medical problems.
  • Applications: Developing artificial organs, replacement limbs, MRI machines, laser systems, and medication delivery devices like insulin pumps or asthma inhalers.

1.1 Arithmetic Sequences

• Mathematical Principles

  • Ordered List: A sequence is an ordered list containing elements or terms labelling by position ($t_1$ is the first term, $n$ is the position, $t_n$ is the nth term).
  • Finite vs. Infinite:
    • Finite: Has a defined end (e.g., $2, 5, 8, 11, 14$).
    • Infinite: Continues indefinitely (e.g., $5, 10, 15, 20, \dots$).
  • Linear Relationship: Arithmetic sequences are discrete linear functions. The common difference ($d$) corresponds to the slope of the graph.

• General Term Formula

  • Formula: $t_n = t_1 + (n - 1)d$
  • Variables:
    • $t_n$: General or nth term.
    • $t_1$: First term (sometimes referred to as $a$).
    • $n$: Number of terms ($n \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers).
    • $d$: Common difference.

• Application Examples

  • Comet Sightings: Edmond Halley predicted the return of the comet (now Halley’s Comet) seen in $1531$, $1607$, and $1682$. It returned in $1758$. The years form an arithmetic sequence with $d \approx 75$ or $76$.
  • Musk-Ox Population: In $1955$, population was $9250$. If it increases by $1650$ per year, the time to reach $100000$ is calculated as:
    • $100000 = 9250 + (n - 1)1650$
    • $n = 56$
    • It would take $56$ years.
  • Staircase Numbers:
    • Two-step staircase: Sum of consecutive columns ($1+2, 2+3, 3+4, \dots$), resulting in $3, 5, 7, \dots$
    • Three-step staircase: Sum of three columns ($1+2+3, 2+3+4, \dots$), resulting in $6, 9, 12, \dots$

1.2 Arithmetic Series

• Gauss’s Discovery

  • History: Carl Friedrich Gauss (born $1777$) calculated the sum of integers from $1$ to $100$ in minutes.
  • Method: Grouping terms into pairs that sum to the same value ($1+100=101, 2+99=101, \dots$). Since there are $100$ terms ($50$ pairs), the sum is $50 \times 101 = 5050$.

• Sum Formulas

  • Standard Formula: $S_n = \frac{n}{2}[2t_1 + (n - 1)d]$
  • Alternative Formula (using last term): $S_n = \frac{n}{2}(t_1 + t_n)$

• Application Examples

  • Firefly Flashes: A firefly flashes $2$ times in minute 1, $4$ in minute 2, and $6$ in minute 3.
    • Flashes in 30th minute: $t_{30} = 2 + (30-1)2 = 60$
    • Total flashes in 30 minutes: $S_{30} = \frac{30}{2}(2 + 60) = 930$
  • Handshakes: In a group of $6$ people, if everyone shakes hands once, it models the series $1 + 2 + 3 + 4 + 5 = 15$ handshakes.
  • Triangular Numbers: Sequence where dots form triangles ($1, 3, 6, 10, \dots$). The nth triangular number is $S_n = \frac{n}{2}(n + 1)$.

1.3 Geometric Sequences

• General Principles

  • Definition: Each term is found by multiplying the previous term by a non-zero constant ratio ($r$).
  • Formula: $t_n = t_1 r^{n-1}$
  • Behavior of r:
    • If $|r| > 1$, the sequence terms grow in magnitude.
    • If $0 < |r| < 1$, the sequence terms decrease in magnitude.
    • If $r < 0$, the terms alternate signs.

• Real-World Models

  • Microbiology: Bacteria reproduction by splitting (binary fission) produces geometric sequences ($2, 4, 8, 16, \dots$ with $r=2$).
  • Acoustics: Piano keyboard frequencies from $A_0$ ($27.5\,Hz$) to $C_8$ ($4186.009\,Hz$) approximate a geometric sequence with $r \approx 1.06$.
  • Optics: Photocopier reductions (e.g., $67\%\,r = 0.67$) follow geometric patterns.
  • One Grain of Rice (Legend): Rani asks the Raja for one grain of rice, doubled daily for $30$ days. Grains on day 30: $t_{30} = 1 \times 2^{29} = 536870912$.

1.4 Geometric Series

• Sum of Geometric Terms

  • Derivation: derived by subtracting the original series ($S_n$) from the series multiplied by the ratio ($rS_n$).
  • Formula 1 (using n): $S_n = \frac{t_1(r^n - 1)}{r - 1}, r \neq 1$
  • Formula 2 (using $t_n$): $S_n = \frac{rt_n - t_1}{r - 1}, r \neq 1$

• Fractals and Geometry

  • Definition: A figure generated by repeating a simple pattern infinitely, exhibiting self-similarity.
  • Fractal Tree: Each stage adds branches (e.g., $1, 2, 4, 8, \dots$).
  • Koch Snowflake: Created by adding equilateral triangles to the sides of an base triangle.
    • Line segments increase by a factor of $4$ each stage.
    • Segment length decreases by a factor of $\frac{1}{3}$.

• Other Applications

  • Fan-out Systems: Communication trees where one person calls $x$ people, who each call $x$ more people.
  • Pharmacology: Drug level calculations ($S_n$ represents the amount remaining after multiple doses, considering the metabolism rate $r$).
  • Sports: Tournament brackets where half the contestants are eliminated each round ($r = \frac{1}{2}$).

1.5 Infinite Geometric Series

• Convergence vs. Divergence

  • Convergent Series: Occurs when $-1 < r < 1$. The sum approaches a finite limit as $n \rightarrow \infty$.
  • Divergent Series: Occurs when $|r| \geq 1$. The sum increases without bounds or oscillates.
  • Zeno’s Paradox: The philosophical argument that motion is impossible because to reach a wall, one must first travel half the distance ($\frac{1}{2}$), then half of that ($\frac{1}{4}$), etc. Mathematically, the sum converges to $1$, resolving the paradox.

• Sum Formula for Infinite Series

  • Formula: $S_{\infty} = \frac{t_1}{1 - r}, |r| < 1$

• Computational Applications

  • Repeating Decimals: Decimals like $0.\overline{584}$ can be written as an infinite series: $0.584 + 0.000584 + 0.000000584 + \dots$
    • $t_1 = 0.584$
    • $r = 0.001$
    • $S_{\infty} = \frac{0.584}{1 - 0.001} = \frac{584}{999}$
  • Oil Depletion: Lifetime production of a well can be estimated if production declines by a constant percentage (e.g., $r = 0.94$).
  • Physics: Calculating the total vertical distance of a bouncing ball.
    • Note: The distance consists of a single initial drop plus an infinite series of double distances (up and down) for every subsequent bounce.

Quick Reference Data and Facts

• Natural Resource Specifics:
  • Gold Production: $62\%$ of the world's discovered gold ($193000\,metric\,tonnes$) is in 4 countries; it would fit in a cube of side length $22\,m$.
  • Silver Production: Total discovered is approximately $1740000\,metric\,tonnes$, fitting in a cube of side length $55\,m$.
  • Materials in Vehicles: An average $1360\,kg$ car contains $110\,kg$ aluminum, $20\,kg$ copper, $10\,kg$ zinc, $113\,kg$ plastics, and $64\,kg$ rubber.
  • Wind Power Performance: In Pincher Creek, a single $660\,kW$ turbine produces $2000\,MW$ annually, enough for $250$ homes, replacing $900000\,kg$ of coal.