Arithmetic Sequence Calculator
Arithmetic Sequence Calculator
Result
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 nแตสฐ 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.
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, โฆ..
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, โฆ
6, 3, 0, -3, -6, -9, -12, -15, โฆ
50, 50.1, 50.2, 50.3, 50.4, 50.5, โฆ
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 nแตสฐ term of any arithmetic sequence:
an = aโ + (n โ 1)d
Where:
an โ the nแตสฐ term of the sequence
d โ the common difference
aโ โ 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
Q1. What are the uses of arithmetic sequences?
Arithmetic sequences are used in various aspects of daily life, including calculating simple interest, straight-line depreciation, accumulating monthly rent, and more. For example, when a person saves money regularly in fixed amounts, the total savings at each point form an arithmetic progression.
Q2. Why is a sequence not arithmetic?
To check whether a sequence is arithmetic, geometric, or neither, we examine its terms. We look for a continuous difference or a continuous ratio. If neither condition is satisfied, the sequence is neither arithmetic nor geometric.
Q3. Where are arithmetic sequences used in real life?
When a person saves money in equal instalments, the total savings after each period form an arithmetic progression. Similarly, if a person drives a car at a constant speed on a highway, the fuel remaining in the tank, recorded every minute, forms another arithmetic progression.
Q4. What pattern does an arithmetic sequence follow?
Simply put, an arithmetic sequence follows a consistent pattern where each term is formed by adding (or subtracting) the same fixed value to the previous term. For example, in the sequence 2, 5, 8, 11, 14โฆ each term is formed by adding 3 to the term before it.
