Arithmetic Series Formulas and Properties

Arithmetic Series

Formula Derivation

• An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term.
• The formula for the nth term of an arithmetic series is: an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.
• The formula can be derived by considering the pattern of the sequence and using mathematical induction.

Sum Of Finite Series

• The formula for the sum of a finite arithmetic series is: Sn = (n/2)(a1 + an), where Sn is the sum of the first n terms, a1 is the first term, and an is the last term.
• This formula can be derived by pairing the first and last terms, the second and second-to-last terms, and so on, and observing that the sum of each pair is equal to a1 + an.
• The formula can also be written as: Sn = (n/2)(2a1 + (n - 1)d), by substituting the formula for the nth term.

Real-world Applications

• Arithmetic series have many real-world applications, such as:
  • Calculating the total cost of a series of payments, where each payment increases by a fixed amount.
  • Modeling population growth, where the population increases by a fixed amount each year.
  • Determining the total distance traveled by an object moving at a constant acceleration.

Infinite Series Convergence

• An infinite arithmetic series converges if the common difference d is negative and the magnitude of d is less than 1.
• The sum of an infinite arithmetic series can be found using the formula: S = a1 / (1 - d), where a1 is the first term and d is the common difference.
• The convergence of an infinite arithmetic series can be determined using the ratio test or the root test.

Recursive Formulae

• A recursive formula for an arithmetic series is a formula that defines each term in terms of previous terms.
• The recursive formula for an arithmetic series is: an = an-1 + d, where an is the nth term and d is the common difference.
• Recursive formulae can be used to calculate the nth term of an arithmetic series, and to prove the formula for the sum of a finite arithmetic series.

Arithmetic Series

Formula Derivation

• An arithmetic series is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term.
• The formula for the nth term is an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.

Sum Of Finite Series

• The formula for the sum of a finite arithmetic series is Sn = (n/2)(a1 + an), where Sn is the sum of the first n terms.
• This formula can be derived by pairing the first and last terms, the second and second-to-last terms, and so on.
• The formula can also be written as Sn = (n/2)(2a1 + (n - 1)d) by substituting the formula for the nth term.

Real-world Applications

• Arithmetic series have many real-world applications, such as:
  • Calculating the total cost of a series of payments, where each payment increases by a fixed amount.
  • Modeling population growth, where the population increases by a fixed amount each year.
  • Determining the total distance traveled by an object moving at a constant acceleration.

Infinite Series Convergence

• An infinite arithmetic series converges if the common difference d is negative and the magnitude of d is less than 1.
• The sum of an infinite arithmetic series can be found using the formula S = a1 / (1 - d).
• The convergence of an infinite arithmetic series can be determined using the ratio test or the root test.

Recursive Formulae

• A recursive formula for an arithmetic series is a formula that defines each term in terms of previous terms.
• The recursive formula for an arithmetic series is an = an-1 + d, where an is the nth term and d is the common difference.
• Recursive formulae can be used to calculate the nth term of an arithmetic series, and to prove the formula for the sum of a finite arithmetic series.