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] ]