Finite Arithmetic Series: Sum and Formula

Sum Of Finite Series

• An arithmetic series is the sum of a finite number of terms in an arithmetic sequence
• The sum of a finite arithmetic series can be calculated using the formula:
  • S = (n/2) × (a + l)
  • Where:
    • S = sum of the series
    • n = number of terms
    • a = first term
    • l = last term

Formula Derivation

• The formula for the sum of a finite arithmetic series can be derived by:
  • Pairing the first and last terms, second and second-to-last terms, and so on
  • Noting that the sum of each pair is equal to the average of the first and last terms multiplied by the number of terms
  • Simplifying the expression to arrive at the formula S = (n/2) × (a + l)

Infinite Series Convergence

• An infinite arithmetic series converges if the common difference is zero (d = 0)
• If the common difference is not zero, the series diverges
• The convergence of an infinite arithmetic series can be determined using the following rules:
  • If the series is finite, it converges
  • If the series is infinite and the common difference is zero, it converges
  • If the series is infinite and the common difference is not zero, it diverges

Real-world Applications

• Arithmetic series have numerous real-world applications, including:
  • Calculating the total cost of a series of payments
  • Determining the total distance traveled by an object moving with a constant acceleration
  • Modeling population growth or decline
  • Analyzing financial data, such as investment returns or depreciation

Recursive Formulae

• A recursive formula is a formula that defines each term in an arithmetic sequence using the previous term
• The recursive formula for an arithmetic sequence is:
  • an = an-1 + d
  • Where:
    • an = nth term
    • an-1 = (n-1)th term
    • d = common difference
• Recursive formulae can be used to calculate individual terms in an arithmetic sequence or to find the sum of a finite series.

Arithmetic Series

• An arithmetic series is the sum of a finite number of terms in an arithmetic sequence.

Formula for Sum of Arithmetic Series

• The sum of a finite arithmetic series can be calculated using the formula: S = (n/2) × (a + l)
• Where:
  • S = sum of the series
  • n = number of terms
  • a = first term
  • l = last term

Derivation of Formula

• The formula can be derived by pairing the first and last terms, second and second-to-last terms, and so on.
• The sum of each pair is equal to the average of the first and last terms multiplied by the number of terms.
• The expression can be simplified to arrive at the formula S = (n/2) × (a + l)

Convergence of Infinite Arithmetic Series

• An infinite arithmetic series converges if the common difference is zero (d = 0).
• If the common difference is not zero, the series diverges.
• The convergence of an infinite arithmetic series can be determined using the following rules:
  • If the series is finite, it converges.
  • If the series is infinite and the common difference is zero, it converges.
  • If the series is infinite and the common difference is not zero, it diverges.

Real-world Applications of Arithmetic Series

• Arithmetic series have numerous real-world applications, including:
  • Calculating the total cost of a series of payments.
  • Determining the total distance traveled by an object moving with a constant acceleration.
  • Modeling population growth or decline.
  • Analyzing financial data, such as investment returns or depreciation.

Recursive Formula for Arithmetic Sequence

• A recursive formula is a formula that defines each term in an arithmetic sequence using the previous term.
• The recursive formula for an arithmetic sequence is: an = an-1 + d
• Where:
  • an = nth term
  • an-1 = (n-1)th term
  • d = common difference
• Recursive formulae can be used to calculate individual terms in an arithmetic sequence or to find the sum of a finite series.