Arithmetic and Geometric Sequences: Formulas, Differences, and Examples

In mathematics, computer science, and even finance, we often deal with lists of numbers that follow a specific pattern. These are called sequences. When we add the terms of a sequence together, we get a series. The two most common and important types of sequences are Arithmetic Sequences and Geometric Sequences. Understanding the formulas behind these progressions helps solve problems ranging from simple counting to complex financial compounding.

1. Arithmetic Sequences: Constant Additions

An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d). For example, in the sequence 5, 8, 11, 14, ..., the common difference is 3.

2. Geometric Sequences: Exponential Multiplication

A geometric sequence (or geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a non-zero number called the common ratio (r). For example, in the sequence 3, 6, 12, 24, ..., the common ratio is 2.

Figure 4: Side-by-side comparison of linear growth (Arithmetic Progression) vs exponential growth (Geometric Progression).

3. Summing It Up: Arithmetic and Geometric Series

A series is the sum of the terms in a sequence. We use summation formulas to find the total sum of the first n terms (represented by S_n) without adding every single number manually.

• Arithmetic Series Sum: S_n = (n / 2) × (a_1 + a_n) — or S_n = (n / 2) × [2a_1 + (n - 1)d]
• Geometric Series Sum: S_n = a_1 × (1 - r^n) / (1 - r) — where r ≠ 1

4. Practical Examples in Progression

• Arithmetic Example: A sequence starts 5, 8, 11, 14... Find the 10th term. Here, the first term (a_1) is 5, and the common difference (d) is 3. Solve: a_10 = 5 + (10 - 1) × 3 = 5 + 27 = 32.
• Geometric Example: A sequence starts 3, 6, 12, 24... Find the 8th term. Here, the first term (a_1) is 3, and the common ratio (r) is 2. Solve: a_8 = 3 × 2^(8 - 1) = 3 × 128 = 384.
• Real-World Compounding: Compound interest is a geometric progression. If you invest $1,000 at a 10% annual return, your balance year-by-year ($1,000, $1,100, $1,210, $1,331...) forms a geometric sequence with a common ratio of 1.10.

Ready to calculate progressions? Scroll down to the interactive Sequence & Progression Calculator below, input your parameters, and find the nth term and series sums instantly with step-by-step arithmetic and geometric results!