Arithmetic Sequences

Definition

• An arithmetic sequence is a sequence where each term is obtained by adding a constant (called the common difference, denoted as d) to the previous term.
• Formula for the nth term of an arithmetic sequence:
  • [ a_n = a_1 + (n-1)d ]
  • Where:
    • ( a_n ) = nth term
    • ( a_1 ) = first term
    • ( n ) = term number
    • ( d ) = common difference

Finding the Common Difference

Example 1: Sequence 2, 5, 8, 11

• First term (a1): 2
• Common difference (d): 3 (5-2, 8-5, 11-8)
• Next term: 14 (11 + 3)

Example 2: Sequence 1, 1/3, 1/3, 2/3, 2

• First term (a1): 1
• Common difference (d): 1/3 (2/3 - 1/3)

Example 3: Sequence 17, 12, 7, 2, -3

• First term (a1): 17
• Common difference (d): -5 (12 - 17)
• Next term: -8 (-3 - 5)

Deriving the nth Term

• General form: ( a_n = a_1 + (n-1)d )
• Example: Find the 16th term of the sequence 1, 5, 9, 13
  • First term (a1): 1
  • Common difference (d): 4 (5-1)
  • n: 16
  • Calculation: ( a_{16} = 1 + (16-1)4 = 1 + 60 = 61 )

Further Examples

Example: Find the 20th term of: 25, 23, 21, 19, 17

• First term (a1): 25
• Common difference (d): -2
• Calculation: ( a_{20} = 25 + (20-1)(-2) = 25 - 38 = -13 )

Example: Find the nth term for sequence 3, 7, 11, 15

• Common difference (d): 4
• nth term: ( a_n = 3 + (n-1)4 = 4n - 1 )

Example: Determine which term is 5 in the sequence 50, 45, 40, 35

• First term (a1): 50
• Common difference (d): -5
• Set the equation: ( 5 = 50 + (n-1)(-5) )
• Solve for n: ( 5n = 50 \Rightarrow n = 10 )

Example: Find nth term where a4 = 34 and a10 = 22

• Set equations:
  • ( a_4 = a_1 + 3d = 34 )
  • ( a_{10} = a_1 + 9d = 22 )
• Solve for a1 and d:
  • Common difference (d) = -2, First term (a1) = 40
  • nth term: ( a_n = 40 + (n-1)(-2) = -2n + 42 )

Real-life Application: Salary Calculation

• Jeffrey's starting salary: 240,000 with annual raises of 20,000.
• Calculate salary for 10th year:
  • Calculation: ( 240000 + (10-1)20000 = 240000 + 180000 = 420000 )

Conclusion

• Arithmetic sequences are defined by a starting term and a common difference.
• The nth term can be calculated using the general formula and can be applied in various real-life situations.