Arithmetic Progression Definition and Properties

Definition and Properties

• Arithmetic Progression (AP): A sequence of numbers in which the difference between consecutive terms is constant.

  • General form: ( a, a + d, a + 2d, a + 3d, \ldots )
  • Where:
    • ( a ) = first term
    • ( d ) = common difference

• Properties:

  • The ( n )-th term can be expressed as ( a_n = a + (n - 1)d ).
  • The terms are uniformly spaced; thus, the average of the first and last term is equal to the average of all terms.
  • If ( n ) is odd, the middle term is the average of the first and last term.
  • The sum of the first ( n ) terms can be calculated using specific formulas.

Formulas For Nth Term

• Nth Term Formula:
  • ( a_n = a + (n - 1)d )
    • ( a_n ) = n-th term
    • ( a ) = first term
    • ( d ) = common difference
    • ( n ) = term number

Sum Of First N Terms

• Sum Formula:

  • ( S_n = \frac{n}{2} [2a + (n - 1)d] ) (when ( n ) is known)
  • Alternatively: ( S_n = \frac{n}{2} [a + a_n] ) (using the first and n-th term)
    • ( S_n ) = sum of the first ( n ) terms
    • ( a ) = first term
    • ( a_n ) = n-th term
    • ( d ) = common difference
    • ( n ) = number of terms

• Special Cases:

  • If ( d = 0 ) (constant sequence), each term is equal to ( a ), and ( S_n = n \cdot a ).
  • If ( n ) is large, the sum can be approximated in different contexts (e.g., for series analysis).

Arithmetic Progression (AP)

• A sequence where the difference between consecutive terms remains constant.
• General form: ( a, a + d, a + 2d, a + 3d, \ldots )
  • ( a ): first term
  • ( d ): common difference

Key Properties of AP

• The ( n )-th term can be calculated using ( a_n = a + (n - 1)d ).
• Terms are uniformly spaced, making the average of the first and last term equal to the average of all terms in the sequence.
• When ( n ) is odd, the middle term represents the average of the first and last terms.
• The sum of the first ( n ) terms can be determined using specific formulas.

Nth Term Formula

• Formula for the ( n )-th term: ( a_n = a + (n - 1)d )
  • ( a_n ): value of the n-th term
  • ( a ): first term of the sequence
  • ( d ): common difference
  • ( n ): position of the term in the sequence

Sum of First N Terms

• Sum formula when ( n ) is given: ( S_n = \frac{n}{2} [2a + (n - 1)d] ).
• An alternative sum formula is ( S_n = \frac{n}{2} [a + a_n] ), utilizing the first and n-th terms.
  • ( S_n ): total of the first ( n ) terms
  • ( a ): first term
  • ( a_n ): value of the n-th term
  • ( d ): common difference
  • ( n ): total number of terms

Special Cases in AP

• If ( d = 0 ), all terms in the sequence are equal to ( a ), resulting in ( S_n = n \cdot a ).
• For large ( n ), the sum may be approximated depending on the context (e.g., in convergence analysis or series).

This quiz covers the essential definitions and properties of Arithmetic Progression (AP). It includes the general form, formulas for the n-th term, and the sum of the first n terms. Test your understanding of these concepts and their applications.