Arithmetic Sequence Calculator

This arithmetic sequence calculator, also known as an arithmetic series calculator, is a simple tool for exploring sequences of numbers formed by repeatedly adding a fixed value. You can use it to determine any aspect of a sequence – whether it’s the first term, the common difference, the nᵗʰ term, or the sum of the first n terms. Jump right in to use it or keep reading to understand how it works.

In this guide, we define what an arithmetic sequence is, explain the sequence formula that powers the calculator, and provide the equation for calculating an arithmetic series (the sum of an arithmetic progression). We also highlight the key differences between arithmetic and geometric sequences and provide a clear, simple example that shows how to apply this tool effectively.

What is an Arithmetic Sequence?

To understand an arithmetic sequence, you first need to know what a sequence is. In mathematics, a sequence is an ordered collection of objects, such as numbers or letters. Each object in a sequence is called a term or element. Often, the same object can appear multiple times in a sequence.

An arithmetic sequence is a specific type of sequence made up of numbers. Each number after the first is formed by adding a fixed value, called the common difference, to the previous number. It can be either a simple number if it has a finite number of terms (for example, 20 terms), or it can be completely arbitrary if the number of terms is not finite.

Every arithmetic sequence is completely determined by two values: the first term and the common difference. Knowing this allows you to write the entire sequence.

How to Use Arithmetic Sequence Calculator?

Follow the simple steps given below to calculate the terms in an arithmetic progression using the Arithmetic Sequence Calculator:

Step 1: Open QMath’s online Arithmetic Sequence Calculator.
Step 2: Enter the first term (a) and the common difference (d) in the designated fields of the Arithmetic Sequence Calculator.
Step 3: Press the “Find” button to generate the terms of the arithmetic sequence.
Step 4: Press the “Reset” button to clear the inputs and provide new values.

Can you identify the common difference in each sequence? Hint: Subtract any term from the term that follows it.

From these examples, you can see that the common difference does not have to be a whole number – it can be a fraction or even a negative number.

Arithmetic Sequence Calculator: Example of Use

Let’s look at a clear example that can be solved using the arithmetic sequence formula. We will examine a familiar physics situation – free fall.

A stone falls freely into a deep shaft. In the first second, it falls 4 meters. With each subsequent second, the distance traveled increases by 9.8 meters. How much distance does the stone travel in total from the fifth to the ninth second?

The distances form an arithmetic sequence with the first term a = 4 meters and the constant difference d = 9.8 meters.

First, we calculate the total distance covered by the stone by finding the partial sum of S₉ (where n = 9):

S₉ = n/2 × [2a₁ + (n − 1)d]
S₉ = 9/2 × [2 × 4 + (9 − 1) × 9.8] = 388.8 meters

So, during the first nine seconds, the stone covers a total of 388.8 meters. But our goal is to find the distance covered only from the fifth to the ninth second. To do that, we subtract the distance covered in the first four seconds (S₄) from S₉.

S₄ = n/2 × [2a₁ + (n − 1)d]
S₄ = 4/2 × [2 × 4 + (4 − 1) × 9.8] = 74.8 meters

The value of S₄ is 74.8 meters. Now let’s subtract:

Distance = S₉ − S₄ = 388.8 − 74.8 = 314 meters

There is another way to solve this problem. You can calculate the distance covered during the fifth, sixth, seventh, eighth, and ninth seconds using the arithmetic progression formula, then add them together. If you try this approach, you will find that it gives the same final result.

How Does an Arithmetic Progression Calculator Work?

An arithmetic progression (AP) refers to a series where the difference between each successive term remains constant. In AP, you find new terms by adding a fixed value to the previous term. Mathematics includes several types of progressions such as geometric and harmonic progressions. The terms of AP follow the pattern shown below:

AP = a, a + d, a + 2d, a + 3d, a + 4d, …..

Arithmetic Sequence Definition and Nomenclature

When exploring arithmetic sequences, naming conventions can sometimes cause confusion. Two terms you will see frequently are arithmetic sequence and series. An arithmetic sequence is also known as an arithmetic progression, while a series is often called a partial sum.

Examples of Arithmetic Sequences

Here are some examples of arithmetic sequences:

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, …

6, 3, 0, -3, -6, -9, -12, -15, …

50, 50.1, 50.2, 50.3, 50.4, 50.5, …

Arithmetic Sequence Formula

Suppose you want to find the 30th term of one of the above sequences. It would take time to write out all 30 terms. Instead, you just need to add the common difference to the first term 29 times.

We can generalize this to the formula for the nᵗʰ term of any arithmetic sequence:

an = a₁ + (n − 1)d

Where:

an — the nᵗʰ term of the sequence

d — the common difference

a₁ — the first term of the sequence

Arithmetic Series to Infinity

When summing an arithmetic sequence, you usually choose a value for n to find the partial sum. But what if you want to sum all the terms indefinitely?

If the sequence is infinite, the total sum will also grow without limit, whether the common difference is positive, negative, or zero. However, this is not true for every type of sequence. For example, a geometric sequence can have a finite sum, even if it extends to infinity.

FAQs: Frequently Asked Questions