Deriving the Formula for the Sum of an Arithmetic Series

Jun 4, 2024

Deriving the Formula for the Sum of an Arithmetic Series

Introduction

  • Arithmetic Series: The sum of a sequence where each consecutive term has a common difference.
  • Example: Series with common difference of 1: 1, 2, 3, ... , 100.
  • Goal: Derive a general formula for the sum of an arithmetic series.

Example with Specific Numbers

  1. Given Series: 1, 2, 3, ..., 100.

  2. Method: Write the series in reverse and add it to the original series.

    Steps:

    • Original: S = 1 + 2 + 3 + ... + 100
    • Reverse: S = 100 + 99 + 98 + ... + 1
    • Adding these: (1+100) + (2+99) + (3+98) + ...
    • Each paired sum = 101
    • Number of terms = 100
  3. Calculation:

    • S + S = 100 * 101
    • 2S = 100 * 101
    • S = 100 * 101 / 2 = 5050

Generalizing the Formula

  1. Define Variables:

    • a = first term
    • d = common difference
    • n = number of terms
  2. Series Representation:

    • Sₙ = a + (a + d) + (a + 2d) + ... + [a + (n-1)d]
  3. Use the same summing trick:

    • Write the series in reverse.
    • Add it to the original series.

    Steps:

    • Original: Sₙ = a + (a + d) + (a + 2d) + ... + [a + (n-1)d]
    • Reverse: Sₙ = [a + (n-1)d] + [a + (n-2)d] + ... + a
    • Adding these: Each sum = 2a + (n-1)d
    • Number of terms = n
  4. Derive the Formula:

    • 2Sₙ = n * [2a + (n-1)d]
    • Sₙ = n/2 * [2a + (n-1)d]

Final Formula

  • Sum of Arithmetic Series:

    [ Sₙ = \frac{n}{2} [2a + (n-1)d] ]