Arithmetic Sequences Quiz

Arithmetic Sequences Basics

• An arithmetic sequence is defined by a first term and a common difference, with each term computed by adding the common difference to the previous term.
• The nth term of an arithmetic sequence can be calculated using the formula:
  • ( a_n = a_1 + (n-1)d )
  • Where ( a_n ) is the nth term, ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the position of the term.

Example Calculations

• 10th Term Calculation

  • First term: 5
  • Common difference: 3
  • 10th term:
    • ( a_{10} = 5 + (10-1) \times 3 = 5 + 27 = 32 )

• Common Difference Calculation

  • First term: 12
  • Fifth term: 24
  • Common difference:
    • Since the fifth term can be expressed as ( a_5 = a_1 + (5-1)d ):
    • ( 24 = 12 + 4d )
    • Rearranging gives ( 4d = 12 ), thus ( d = 3 )

• 15th Term Calculation with Negative Difference

  • First term: 4
  • Common difference: -2
  • 15th term:
    • ( a_{15} = 4 + (15-1)(-2) = 4 - 28 = -24 )

• Finding Common Difference from Terms

  • First term: 7
  • Seventh term: 35
  • Expression setup:
    • ( a_7 = a_1 + (7-1)d ) translates to ( 35 = 7 + 6d )
    • Solving yields ( 6d = 28 ), resulting in ( d = \frac{14}{3} ) or approximately 4.67

• 20th Term Calculation

  • First term: -10
  • Common difference: 5
  • 20th term:
    • ( a_{20} = -10 + (20-1)5 = -10 + 95 = 85 )