Arithmetic Sequences Overview

Arithmetic Sequence

• Definition: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

• Common Difference (d):

  • The constant difference between consecutive terms.
  • Calculated as ( d = a_{n} - a_{n-1} ).

• General Form:

  • The n-th term of an arithmetic sequence can be expressed as: [ a_n = a_1 + (n-1) \cdot d ]
  • Where:
    • ( a_n ) = n-th term
    • ( a_1 ) = first term
    • ( n ) = term number
    • ( d ) = common difference

• Examples:

  • Sequence: 2, 5, 8, 11, ...
    • Here, ( a_1 = 2 ) and ( d = 3 ).
  • Sequence: 10, 7, 4, 1, ...
    • Here, ( a_1 = 10 ) and ( d = -3 ).

• Sum of an Arithmetic Sequence:

  • The sum ( S_n ) of the first n terms can be calculated using: [ S_n = \frac{n}{2} \cdot (a_1 + a_n) ]
  • Alternatively: [ S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d) ]

• Key Properties:

  • Arithmetic sequences can be finite (with a limited number of terms) or infinite (continuing indefinitely).
  • The graph of an arithmetic sequence is a straight line.

• Applications:

  • Used in various fields such as finance (e.g., calculating linear interest), computer science (e.g., algorithm analysis), and everyday problems (e.g., budgeting).

• Identifying an Arithmetic Sequence:

  • Check if the difference between consecutive terms is constant.

Important Notes

• Always verify the common difference to confirm the sequence is arithmetic.
• The first term and common difference are crucial for defining the sequence's behavior.

Arithmetic Sequence

• An arithmetic sequence features a constant difference between consecutive terms, known as the common difference (d).
• The common difference can be computed by subtracting the previous term from the current term, represented as ( d = a_{n} - a_{n-1} ).

General Form

• The n-th term of an arithmetic sequence is determined by: [ a_n = a_1 + (n-1) \cdot d ] where:
  • ( a_n ) = n-th term
  • ( a_1 ) = first term
  • ( n ) = term number
  • ( d ) = common difference

Examples

• Example Sequence: 2, 5, 8, 11...
  • Initial term (( a_1 )) is 2
  • Common difference (( d )) is 3
• Another Example: 10, 7, 4, 1...
  • Initial term (( a_1 )) is 10
  • Common difference (( d )) is -3

Sum of an Arithmetic Sequence

• The sum (( S_n )) of the first n terms can be calculated by: [ S_n = \frac{n}{2} \cdot (a_1 + a_n) ]
• An alternative formula for the sum is: [ S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d) ]

Key Properties

• Arithmetic sequences can be either finite, containing a specific number of terms, or infinite, continuing indefinitely.
• The graphical representation of an arithmetic sequence appears as a straight line, illustrating the linear growth or decline.

Applications

• Commonly utilized in finance for calculating linear interest and in computer science for algorithm analysis.
• Practical in everyday scenarios, such as budgeting and financial planning.

Identifying an Arithmetic Sequence

• Confirm an arithmetic sequence by checking if the difference between consecutive terms remains constant.
• The first term and the common difference are essential for determining the sequence's characteristics and behavior.