What is the sum of a specified number of terms of an arithmetic sequence called?

The sum of a specified number of terms of an arithmetic sequence is called an arithmetic series.

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is known as the common difference. An arithmetic series, on the other hand, is the sum of a specified number of terms in such a sequence.

Understanding Arithmetic Series

The sum of the first n terms of an arithmetic sequence (also called an arithmetic series) can be calculated using a specific formula. Consider the arithmetic sequence: a₁, a₂, a₃, ..., a_n, where a₁ is the first term and a_n is the nth term.

• Definition: As stated in our reference, "The sum of the first n terms of an arithmetic sequence is called an arithmetic series".
• Example: 1 + 3 + 5 + 7 + 9 + ... is an example of an arithmetic series. In this case, it's the sum of terms from the arithmetic sequence 1, 3, 5, 7, 9, ...

Formulas for Arithmetic Series

There are two common formulas to calculate the sum (S_n) of the first n terms of an arithmetic series:

1. Using the first and last terms:

   S_n = n/2 * (a₁ + a_n)

   Where:

   • S_n is the sum of the first n terms
   • n is the number of terms
   • a₁ is the first term
   • a_n is the last (nth) term

2. Using the first term and common difference:

   S_n = n/2 * [2a₁ + (n-1)d]

   Where:

   • S_n is the sum of the first n terms
   • n is the number of terms
   • a₁ is the first term
   • d is the common difference

Examples

Let's calculate the sum of the first 5 terms of the arithmetic sequence 1, 3, 5, 7, 9 using both formulas:

1. Using the first and last terms:

   S₅ = 5/2 (1 + 9) = 5/2 10 = 25

2. Using the first term and common difference:

   S₅ = 5/2 [2(1) + (5-1)2] = 5/2 [2 + 8] = 5/2 * 10 = 25

Both formulas give the same result: the sum of the first 5 terms is 25.