Lecture Notes on Arithmetic Sequences and Equation Formulation

Key Concepts

• Arithmetic Sequence: A sequence of numbers with a constant difference between consecutive terms.
• Common Difference (d): The amount by which each term increases (or decreases) from the previous term.
• Equation for an Arithmetic Sequence: For a sequence with first term ( a_1 ) and common difference ( d ), the ( n^{th} ) term ( a_n ) is given by: [ a_n = a_1 + (n-1) \times d ]

Example Sequences and Their Equations

Sequence: 2, 4, 6, 8, 10

• Common Difference: 2
• Equation: ( a_n = 2n )
• Verification:
  • ( a_1 = 2 \times 1 = 2 )
  • ( a_3 = 2 \times 3 = 6 )
  • ( a_5 = 2 \times 5 = 10 )

Sequence: 3, 6, 9, 12, 15

• Common Difference: 3
• Equation: ( a_n = 3n )
• Verification:
  • ( a_2 = 3 \times 2 = 6 )
  • ( a_4 = 3 \times 4 = 12 )

Sequence: 1, 3, 5, 7, 9

• Common Difference: 2
• Zero Term: Subtract 2 from the first term (1-2 = -1)
• Equation: ( a_n = 2n - 1 )
• Verification:
  • ( a_1 = 2 \times 1 - 1 = 1 )
  • ( a_4 = 2 \times 4 - 1 = 7 )

Sequence: 7, 11, 15, 19

• Common Difference: 4
• Zero Term: 7 - 4 = 3
• Equation: ( a_n = 4n + 3 )
• Verification:
  • ( a_4 = 4 \times 4 + 3 = 19 )

Special Sequences

Alternating Signs: -1, 1, -1, 1

• Equation: ( a_n = (-1)^n )

Alternating Sequence with Reversed Order: 1, -1, 1, -1

• Equation: ( a_n = (-1)^{n-1} )

Perfect Squares: 1, 4, 9, 16, 25

• Equation: ( a_n = n^2 )

Perfect Squares with Alternating Signs

• Equation: ( a_n = n^2 \times (-1)^n )

Fractions in Sequences

• Numerator: Sequence of integers: 1, 2, 3, 4, 5 with equation ( n )
• Denominator: Sequence with common difference 1, starting at 2: ( n + 1 )
• General Equation for Fractions: ( a_n = \frac{n}{n+1} )

Deriving Equations from Sequence Data

Example: Sequence 7, 10, 13, 16, 19

• First Term: 7
• Common Difference: 10 - 7 = 3
• Equation: ( a_n = 3n + 4 )
• Verification:
  • Check the 4th term: ( a_4 = 3 \times 4 + 4 = 16 )

Example: Sequence -8, -6, -4, -2

• First Term: -8
• Common Difference: -6 - (-8) = 2
• Equation: ( a_n = 2n - 10 )
• Verification:
  • Check the 3rd term: ( a_3 = 2 \times 3 - 10 = -4 )

General Approach

• Find the difference between given terms to determine the common difference.
• Use the equation ( a_n = a_1 + (n-1) \times d ) to formulate a general equation.
• Verify the equation with known terms from the sequence.