What is the formula for the sum of an arithmetic series?

The sum of an arithmetic series can be calculated using two primary formulas, depending on the information you have.

Formulas for the Sum of an Arithmetic Series

There are two common formulas used to calculate the sum (Sn) of an arithmetic series:

1. When you know the first term (a1), the last term (an), and the number of terms (n):

   S_n = n/2 (a₁ + a_n)

2. When you know the first term (a1), the number of terms (n), and the common difference (d):

   S_n = n/2 [2a₁ + (n - 1)d]

Explanation of the Variables:

• S_n = Sum of the first n terms
• n = Number of terms in the series
• a₁ = First term in the series
• a_n = Last (nth) term in the series
• d = Common difference between consecutive terms

Example:

Let's say we have an arithmetic series: 2, 4, 6, 8, 10

• a₁ = 2
• d = 2
• n = 5
• a_n = 10

Using the first formula: S₅ = 5/2 (2 + 10) = 5/2 (12) = 30

Using the second formula: S₅ = 5/2 [2(2) + (5 - 1)2] = 5/2 [4 + 8] = 5/2 (12) = 30

Both formulas yield the same result, demonstrating their interchangeability depending on the given information.