How Do You Find the Partial Sum of an Arithmetic Sequence?

You can find the partial sum of an arithmetic sequence using a formula that leverages the number of terms and either the first and last terms or the first term and the common difference.

Understanding Arithmetic Sequences and Partial Sums

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. A partial sum is the sum of a finite number of consecutive terms from an arithmetic sequence.

Formulas for the Partial Sum of an Arithmetic Sequence

There are two primary formulas you can use, depending on the information you have:

1. Using the First and Last Terms:

The formula is:

S_n = n(a₁ + a_n) / 2

Where:

• S_n is the sum of the first n terms (the nth partial sum).
• n is the number of terms you are summing.
• a₁ is the first term of the sequence.
• a_n is the nth term (the last term you are summing).

2. Using the First Term and Common Difference:

The formula is:

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 you are summing.
• a₁ is the first term of the sequence.
• d is the common difference between consecutive terms.

Example

Let's say we have the arithmetic sequence: 2, 5, 8, 11, 14... and we want to find the sum of the first 5 terms.

Using the First and Last Terms:

• n = 5
• a₁ = 2
• a₅ = 14

S₅ = 5(2 + 14) / 2 = 5(16) / 2 = 40

Using the First Term and Common Difference:

• n = 5
• a₁ = 2
• d = 3 (because 5-2 = 3, 8-5 = 3, etc.)

S₅ = 5/2 [2(2) + (5 - 1)3] = 5/2 [4 + 12] = 5/2 [16] = 40

Both methods yield the same result: the sum of the first 5 terms is 40.

Choosing the Right Formula

• Use the first formula (S_n = n(a₁ + a_n) / 2) if you know the first and last terms.
• Use the second formula (S_n = n/2 [2a₁ + (n - 1)d]) if you know the first term and the common difference.

In Summary

Calculating the partial sum of an arithmetic sequence involves using one of two formulas, each requiring slightly different information about the sequence. Understanding these formulas allows you to efficiently calculate the sum of a specified number of terms in any arithmetic sequence.