Arithmetic Series Calculator

First Term (a₁):

Common Difference (d):

Number of Terms (n):

Results:

Nth Term (a_n): –

Sum of Series (S_n): –

function calculateArithmeticSeries() { var firstTerm = parseFloat(document.getElementById('firstTerm').value); var commonDifference = parseFloat(document.getElementById('commonDifference').value); var numberOfTerms = parseInt(document.getElementById('numberOfTerms').value); if (isNaN(firstTerm) || isNaN(commonDifference) || isNaN(numberOfTerms) || numberOfTerms <= 0) { document.getElementById('nthTermResult').innerHTML = 'Nth Term (a_n): Please enter valid numbers for all fields. Number of terms must be positive.'; document.getElementById('sumSeriesResult').innerHTML = 'Sum of Series (S_n): -'; return; } // Calculate the Nth term (a_n = a₁ + (n – 1)d) var nthTerm = firstTerm + (numberOfTerms – 1) * commonDifference; // Calculate the Sum of the Series (S_n = n/2 * (a₁ + a_n)) var sumOfSeries = (numberOfTerms / 2) * (firstTerm + nthTerm); document.getElementById('nthTermResult').innerHTML = 'Nth Term (a_n): ' + nthTerm.toFixed(2); document.getElementById('sumSeriesResult').innerHTML = 'Sum of Series (S_n): ' + sumOfSeries.toFixed(2); }

Understanding the Arithmetic Series Calculator

An arithmetic series is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference. This calculator helps you quickly determine the Nth term and the sum of an arithmetic series given its fundamental properties.

What is an Arithmetic Series?

Imagine a list of numbers where each number is obtained by adding a fixed value to the previous one. For example, 2, 5, 8, 11, 14… Here, the common difference is 3. An arithmetic series is the sum of the terms in an arithmetic sequence.

Key Components:

First Term (a₁): This is the starting number of your sequence. In the example 2, 5, 8…, the first term is 2.
Common Difference (d): This is the constant value added to each term to get the next term. In our example, the common difference is 3. It can be positive, negative, or zero.
Number of Terms (n): This is how many numbers are in your sequence that you want to consider for the Nth term or the sum.
Nth Term (a_n): This is the value of the term at a specific position 'n' in the sequence.
Sum of the Series (S_n): This is the total sum of all the terms from the first term up to the Nth term.

Formulas Used:

The calculator uses two primary formulas to determine the results:

Nth Term (a_n): The formula to find any term in an arithmetic sequence is:
a_n = a₁ + (n - 1)d
Where:
- a_n is the Nth term
- a₁ is the first term
- n is the number of terms
- d is the common difference
Sum of the Series (S_n): The formula to find the sum of the first 'n' terms of an arithmetic series is:
S_n = n/2 * (a₁ + a_n)
Alternatively, if a_n is not known:
S_n = n/2 * (2a₁ + (n - 1)d)
Our calculator first finds a_n and then uses the first sum formula.

How to Use the Calculator:

Enter the First Term (a₁): Input the starting value of your arithmetic sequence.
Enter the Common Difference (d): Input the constant value that is added to each term.
Enter the Number of Terms (n): Specify how many terms you want to include in your calculation. This must be a positive whole number.
Click "Calculate Series": The calculator will instantly display the value of the Nth term and the total sum of the series up to that Nth term.

Example:

Let's say you have an arithmetic series where the first term is 5, the common difference is 3, and you want to find the 7th term and the sum of the first 7 terms.

First Term (a₁): 5
Common Difference (d): 3
Number of Terms (n): 7

Using the formulas:

Nth Term (a_7): a_7 = 5 + (7 - 1) * 3 = 5 + 6 * 3 = 5 + 18 = 23
Sum of Series (S_7): S_7 = 7/2 * (5 + 23) = 3.5 * 28 = 98

Inputting these values into the calculator will yield an Nth Term of 23.00 and a Sum of Series of 98.00.