Sequences and Functions

Number Sequences

Introduction

Number sequences are sets of numbers arranged in a specific order, following a particular rule. These sequences can help us identify patterns and make predictions about future numbers in the series.

Types of Sequences

• Arithmetic Sequences: In an arithmetic sequence, the difference between consecutive terms is constant. This difference is known as the common difference.
  • Example: 2, 5, 8, 11, 14,... (Common difference = 3)
• Geometric Sequences: In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Example: 3, 6, 12, 24, 48,... (Common ratio = 2)
• Fibonacci Sequence: This sequence is characterized by each term being the sum of the two preceding terms.
  • Example: 0, 1, 1, 2, 3, 5, 8, 13,...

Patterns and Rules

Identifying Patterns

Recognizing patterns in sequences involves observing the differences or ratios between terms.
Here are some steps to identify patterns:

• Look at the differences: Check if the sequence is arithmetic by finding the difference between consecutive terms.
• Check for ratios: Determine if the sequence is geometric by calculating the ratio between consecutive terms.
• Use known sequences: Compare the sequence to known patterns, such as the Fibonacci sequence.

Finding the nth Term

To find the nth term of a sequence, use a formula specific to the type of sequence:

• Arithmetic Sequence: an = a₁ + (n - 1)d
  • a_n = nth term
  • a₁ = first term
  • d = common difference
  • n = term number
• Example: Find the 10th term of the sequence 2, 5, 8, 11,...
  • First term (a₁) = 2
  • Common difference (d) = 3
  • a₁₀ = 2 + (10 - 1) × 3 = 2 + 27 = 29
• Geometric Sequence: a_n = a₁ × r ^(n-1)
  • a_n = nth term
  • a₁ = first term
  • r = common ratio
  • n = term number
• Example: Find the 5th term of the sequence 3, 6, 12, 24,...
  • First term (a₁) = 3
  • Common ratio (r) = 2
  • a₅ = 3 × 2 ^(5-1) = 3 × 16 = 48

Graphs of Functions

Understanding Functions

A function is a relationship between two sets of numbers where each input (x-value) has exactly one output (y-value). Functions can be represented as equations, tables, or graphs.

Graphing Linear Functions

Linear functions are functions of the form y = mx+b, where:

• m is the slope (rate of change)
• b is the y-intercept (the point where the line crosses the y-axis)

Steps to Graph a Linear Function:

• Identify the slope and y-intercept: From the equation y = 2x + 3, the slope (m) is 2, and the y-intercept (b) is 3.
• Plot the y-intercept: Start at the point (0, 3) on the graph.
• Use the slope: From (0, 3), move up 2 units and right 1 unit to plot the next point.
• Draw the line: Connect the points with a straight line.

Example: Graph y = -x + 2

• Slope (m) = -1
• Y-intercept (b) = 2
• Start at (0, 2)
• Move down 1 unit and right 1 unit to (1, 1)
• Draw the line through the points (0, 2) and (1, 1)

Graphs of Non-linear Functions

Non-linear functions, such as quadratic functions (y = ax² + bx + c), have graphs that are not straight lines.
Example: Graph y = x² - 4

• Create a table of values for x and y
• Plot the points (e.g., (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0))
• Draw a smooth curve through the points

Sequences and Functions

Connecting Sequences to Functions

Sequences can be described by functions. For example, the nth term of an arithmetic sequence can be expressed as a linear function of n.

Example: For the sequence 3, 7, 11, 15,...

• The nth term can be written as a_n = 4n - 1
• This is a linear function where the sequence values are outputs for corresponding inputs of n.