Sum of First N Terms of Arithmetic Sequence Calculator

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This calculator helps you find the sum of the first n terms of such a sequence using the standard arithmetic series formula.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is called the common difference (d).

For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence where the first term (a₁) is 2 and the common difference (d) is 3.

Arithmetic sequences are fundamental in mathematics and have applications in various fields including finance, physics, and computer science.

Formula for Sum of First n Terms

The sum of the first n terms (Sₙ) of an arithmetic sequence can be calculated using the following formula:

Sₙ = n/2 × (2a₁ + (n - 1)d)

Where:

Sₙ = Sum of the first n terms
n = Number of terms
a₁ = First term of the sequence
d = Common difference between terms

This formula is derived from the fact that the sum of an arithmetic sequence is the average of the first and last terms multiplied by the number of terms.

How to Use the Calculator

Enter the first term (a₁) of the arithmetic sequence.
Enter the common difference (d) between terms.
Enter the number of terms (n) you want to sum.
Click the "Calculate" button to compute the sum.
Review the result and chart showing the sequence terms.

The calculator will display the sum of the first n terms along with a chart visualizing the sequence terms.

Worked Examples

Example 1: Simple Sequence

Find the sum of the first 5 terms of the arithmetic sequence where the first term is 3 and the common difference is 2.

S₅ = 5/2 × (2×3 + (5 - 1)×2) = 5/2 × (6 + 8) = 5/2 × 14 = 35

The sum of the first 5 terms is 35.

Example 2: Negative Difference

Find the sum of the first 4 terms of the arithmetic sequence where the first term is 10 and the common difference is -3.

S₄ = 4/2 × (2×10 + (4 - 1)×-3) = 2 × (20 - 9) = 2 × 11 = 22

The sum of the first 4 terms is 22.

FAQ

What is the difference between arithmetic and geometric sequences?

An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms. For example, 2, 4, 6, 8 is arithmetic (d=2), while 2, 4, 8, 16 is geometric (r=2).

Can the common difference be negative?

Yes, the common difference can be negative, which results in a decreasing arithmetic sequence. For example, 10, 7, 4, 1 has a common difference of -3.

What if the number of terms is not an integer?

The formula works for any positive real number of terms, not just integers. However, in practical applications, the number of terms is typically a whole number.