Arithmetic Sequence Calculator - Sum of Arithmetic Series

The arithmetic sequence calculator finds the nth term and the sum of the sequences of all the numbers with the common difference “d”. The arithmetic series solver can solve the arithmetic sequence up to the nth term created by adding a constant value.

What Is An Arithmetic Sequence?

In mathematics, an Arithmetic series is defined as:

“A particular sequence is an arithmetic sequence of numbers where each number is equal to the previous number, plus a constant value called “(d)”.

The existing difference among terms can either be positive or negative depending upon the arithmetic series pattern.

Arithmetic Sequence Formula:

Arithmetic sequence formula is:

\(a_n=a_n1+(n-1) d\)

\(a_n\) = nth term in the given sequence
\(a_1\) = it represents the first term

An arithmetic sequence equation can be simplified and found by using this formula. The arithmetic equation calculates the sum of all numbers from the first to the nth term of the arithmetic sequence

Related: Get the assistance of Harmonic Means Calculator that allows you to calculate the harmonic mean from the dataset, by dividing the sum of reciprocals of the dataset.

How To Calculate Arithmetic Sequence?

Let us resolve a couple of examples to clarify your concept!

Example 1:

If the given sequence is 3, 8, 13, 18, 23, 28, 33, 38,…., then what will be the 9^th term?

Solution:

N = The difference among the numbers in the above sequence is 5
D = first term is 3

We will apply the arithmetic sum formula to further proceed with the calculations:

\(X_n = a_1 + d(n−1) = 3 + 5(n−1)\)

\(3 + 5n − 5\)

\(5n − 2\)

So the next term in the above sequence will be:

\(x_9 = 5×9 − 2\)

\(=43\)

Example 2:

If the term is 1, 4, 7, 10, 13, …, then by applying the formula we can find the unknown number.

Solution:

First term = 1
The common difference = 3
Terms to add up = 10

Therefore, by applying the sum of arithmetic sequence formula and putting values in it: So:

\(Σ^{η – 1}_{k = 0}(α + kd) = n/2 (2a + (η – 1)d)\)

\(Σ^{10 – 1}_{k = 0}(1 + k . 3) = 10/ 2 (2.1+(10 – 1).3)\)

On simplifying we will get: \( 5 (2 + 9·3) = 5 (29) = 145 \)

However, you can get these values by arithmetic sequence calculator directly just by substituting the values in related fields.

Arithmetic Series to Infinity:

While looking for a sum of an arithmetic sequence, it becomes essential to pick the value of “n” to calculate the partial sum. When you want to take the sum of all terms of the sequence, then it will be the sum of infinite numbers. The finite arithmetic series is the sequence of the arithmetic series up to the infinite number.

Instinctively, this sum of infinite numbers will be equal to infinity even if the common difference is positive, negative, or even equal to zero. The case might be different for different types of sequences for example in the case of a geometric sequence, the sum to infinity might be a finite term.

Related: Give the Geometric Mean Calculator a try that helps to calculate the geometric mean for a given data set of numbers or percentages.

How Does The Calculator Work?

The sum of the arithmetic sequence is simple to find through the online tool:

Let’s see how to find the arithmetic sequence!

Input:

Enter the first number of the series and
Enter the common difference
Enter the nth number

Output:

The arithmetic sequence and the nth value
The sum of the arithmetic sequence

Frequently Asked Questions:

What Are The Two Types of Equations In An Arithmetic Sequence?

Arithmetic equations are of two types that include:

Recursive formulas
Explicit formulas

The sum of arithmetic series can be calculated by the Recursive and the Explicit formulas. The arithmetic series sum calculator provides the sum of all the terms in the sequence.

What are Some of the Common Types of Sequences?

There are four basic and common types of sequence:

Arithmetic Sequences that can be solved by implementing an arithmetic sequence formula
Geometric Sequences.
Harmonic Sequences.
Fibonacci Numbers.

Reference:

Wikipedia: Arithmetic progression, Derivation, Standard deviation, Intersections, Examples, and notation, Defining a sequence by recursion.

ChilliMath: Increasing and Decreasing Arithmetic Sequences, examples of Arithmetic Sequence.