Patterns, Sequences, and Graphing

Sequences

• A sequence is a set of numbers in a particular order.

Types of Patterns

• Linear: Example - 3, 6, 9, 12
  • First difference is constant.
  • Produces a straight line when graphed.
  • Also called an arithmetic sequence.
• Quadratic: Example - 1, 6, 15, 28, 45
  • Second difference is constant.
  • Produces a curve when graphed.
• Exponential: Example - 2, 4, 8, 16
  • Each term is multiplied by a particular number to find the next term.
  • Used for exponential growth or decay.
  • Graph increases or decreases rapidly.

Solving Pattern Problems

• Linear Pattern: First difference is constant.
  • E.g., 1st Difference: +3, +3, +3
  • Add the constant difference to find the next term.
• Quadratic Pattern: First difference varies, second difference is constant.
  • E.g., 1st Difference: +5, +9... and 2nd Difference: +4, +4...
  • Use the second difference to identify quadratic sequences.

Finding the nth Term of Quadratic Sequences

• Formula: (T_n = an^2 + bn + c)
  • Use simultaneous equations to solve for (b) and (c).
• Coefficient of (n^2) is half the second difference.

Repeating Patterns

• Patterns can repeat in blocks.
  • E.g., A, B, C, D repeats every 4 terms.
  • Use division to find the remainder for specific term positions.

Graphing

• Represents relationship between two variables.
• Variables on horizontal and vertical axes.
• Slope: Relation between rise and run (Vertical Change/Horizontal Change).
  • Example: Slope = Distance/Time, equivalent to speed.

Analyzing Slope

• Steeper line indicates greater speed.
• Horizontal line indicates no change (speed = 0).
  • Increasing slope: speed increases.
  • Decreasing slope: speed decreases.
  • Constant slope: constant speed.

Drawing Graphs

• Identify two points to draw a line.
• The intersection of lines identifies solutions to equations.
• Example: If data is zero, cost is zero.