Free Arithmetic Sequence Calculator

What Is an Arithmetic Progression?

An arithmetic progression (AP)—often called an arithmetic sequence—is a list of numbers in which every new term is generated by adding a fixed constant to the previous term. This constant is referred to as the common difference ($d$). The first term of the sequence is denoted $a_1$. Knowing these two values allows you to reconstruct the entire progression.

The AP pattern appears in many everyday contexts, such as saving a fixed amount each month, calculating travel distances under constant acceleration, or analyzing evenly spaced data.

Core Formulas

Two key formulas describe every arithmetic progression.

The $n$th Term

The value of any term can be found using:

$$a_n = a_1 + (n-1)d$$

To find the 30th term of a sequence that starts with 5 and has a common difference of 3, for example, you would compute $5 + 29 \times 3 = 92$.

Sum of the First $n$ Terms (Arithmetic Series)

When you need the total of the first $n$ terms, use one of these equivalent expressions:

$$S_n = \frac{n}{2}(a_1 + a_n) \qquad\text{or}\qquad S_n = \frac{n}{2}\bigl[2a_1 + (n-1)d\bigr]$$

The first form uses the last term, while the second only requires the first term and the common difference.

Properties of the Common Difference

The sign and magnitude of $d$ determine the behaviour of the sequence:

• If $d > 0$ → the sequence increases (e.g., $2, 5, 8, 11, \dots$).
• If $d < 0$ → the sequence decreases (e.g., $50, 45, 40, 35, \dots$).
• If $d = 0$ → all terms are identical (a constant sequence).

The common difference can be any rational number—integer, fraction, or decimal—and it is not limited to positive values.

Example Sequences

• $3, 7, 11, 15, 19, \dots$ ( $a_1=3,\ d=4$ )
• $20, 18, 16, 14, \dots$ ( $a_1=20,\ d=-2$ )
• $1.5, 2.0, 2.5, 3.0, \dots$ ( $a_1=1.5,\ d=0.5$ )

Sequence vs. Series

It is important to distinguish between the terms sequence and series:

• An arithmetic sequence is the ordered set of numbers.
• An arithmetic series is the sum of some (or all) of those numbers.

For the sequence $2, 4, 6, 8, 10$, the series $S_3$ would be $2+4+6 = 12$. The tool described here functions as both an arithmetic sequence calculator and an arithmetic series calculator, so you can obtain the sum with a single click.

Arithmetic vs. Geometric Progressions

In a geometric progression (GP) each term is obtained by multiplying the previous one by a constant called the common ratio ($r$). The table below highlights the contrast:

Type	Operation	Example	Key Parameter
Arithmetic	Add $d$	$1, 4, 7, 10, \dots$	Common difference ($d$)
Geometric	Multiply by $r$	$1, 3, 9, 27, \dots$	Common ratio ($r$)

A notable hybrid is the arithmetico‑geometric sequence, formed by multiplying corresponding terms of an AP and a GP.

Not every pattern‑based sequence is arithmetic. The Fibonacci sequence ($0, 1, 1, 2, 3, 5, 8, \dots$) is generated by adding the two previous numbers, not by a fixed difference, and therefore falls outside the scope of this calculator.

Real‑World Application: Free Fall

A stone is released from rest into a deep vertical shaft. During the first second it falls $4\ \text{m}$, and each subsequent second it travels $9.8\ \text{m}$ farther than the previous second. This motion forms an arithmetic progression with $a_1 = 4\ \text{m}$ and $d = 9.8\ \text{m}$.

To find the total distance covered between the 5th and 9th seconds, subtract the sum of the first four seconds ($S_4$) from the sum of the first nine seconds ($S_9$):

$$S_9 = \frac{9}{2}\bigl[2 \times 4 + (9-1) \times 9.8\bigr] = \frac{9}{2} \times (8 + 78.4) = 388.8\ \text{m}$$ $$S_4 = \frac{4}{2}\bigl[2 \times 4 + (4-1) \times 9.8\bigr] = 2 \times (8 + 29.4) = 74.8\ \text{m}$$ $$\text{Distance} = 388.8 - 74.8 = 314\ \text{m}$$

The same result can be obtained by summing the individual terms from the 5th through the 9th term directly.

How to Work with the Calculator

This free arithmetic progression calculator online simplifies every step of the analysis:

• Enter known values – Provide the first term and common difference, or supply any two terms (for example, the 3rd and 7th). The tool will automatically solve for the unknown parameters.
• Compute the $n$th term – Use the built‑in nth term calculator to quickly find any term in the sequence.
• Obtain the sum – The sum of arithmetic sequence feature calculates the total of any initial number of terms using the formulas above.
• Find the common difference – If you have partial information, the common difference calculator deduces $d$ from the given terms.
• Generate a list – Request a range of terms to see the full progression.

The interface is straightforward: fill in the values you know and click to get the missing ones. Whether you need a quick homework check or a thorough exploration, this arithmetic progression calculator delivers reliable results in seconds.