Arithmetic Sequence Calculator

Arithmetic Sequence Calculator

function calculateArithmeticSequence() { var firstTermInput = document.getElementById("firstTerm").value; var commonDifferenceInput = document.getElementById("commonDifference").value; var numberOfTermsInput = document.getElementById("numberOfTerms").value; var a1 = parseFloat(firstTermInput); var d = parseFloat(commonDifferenceInput); var n = parseInt(numberOfTermsInput); if (isNaN(a1) || isNaN(d) || isNaN(n) || n <= 0) { document.getElementById("nthTermResult").innerText = "Please enter valid numbers for all fields. Number of terms must be a positive integer."; document.getElementById("sumOfNTermsResult").innerText = ""; document.getElementById("sequenceListResult").innerText = ""; return; } // Calculate the nth term: a_n = a_1 + (n – 1)d var an = a1 + (n – 1) * d; // Calculate the sum of the first n terms: S_n = n/2 * (2a_1 + (n – 1)d) var Sn = (n / 2) * (2 * a1 + (n – 1) * d); // Generate the first few terms of the sequence (up to 10 or n, whichever is smaller) var sequenceTerms = []; var limit = Math.min(n, 10); // Display up to 10 terms or 'n' terms if n < 10 for (var i = 0; i < limit; i++) { sequenceTerms.push(a1 + i * d); } document.getElementById("nthTermResult").innerText = an.toFixed(2); document.getElementById("sumOfNTermsResult").innerText = Sn.toFixed(2); document.getElementById("sequenceListResult").innerText = sequenceTerms.join(", "); }

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is known as the common difference, often denoted by 'd'. Each term in the sequence is obtained by adding the common difference to the preceding term.

Key Components of an Arithmetic Sequence:

• First Term (a₁): This is the starting number of the sequence.
• Common Difference (d): The constant value added to each term to get the next term. It can be positive, negative, or zero.
• Nth Term (a_n): The value of any specific term at position 'n' in the sequence.
• Sum of First N Terms (S_n): The total sum of all terms from the first term up to the nth term.

Formulas Used in Arithmetic Sequences:

The calculator above uses the following fundamental formulas to determine the properties of an arithmetic sequence:

1. Formula for the Nth Term (a_n):
   an = a₁ + (n - 1)d
   Where:
   • an is the nth term
   • a₁ is the first term
   • n is the term number (position in the sequence)
   • d is the common difference
2. Formula for the Sum of the First N Terms (S_n):
   Sn = n/2 * (2a₁ + (n - 1)d)
   Alternatively, if you know the nth term (a_n):
   Sn = n/2 * (a₁ + an)

How to Use the Arithmetic Sequence Calculator:

Our calculator simplifies the process of finding specific terms and sums within an arithmetic sequence. Here's how to use it:

1. Enter the First Term (a₁): Input the starting value of your sequence.
2. Enter the Common Difference (d): Input the constant value that is added to each term to get the next.
3. Enter the Number of Terms (n): Specify which term number you are interested in (e.g., 10 for the 10th term) and for how many terms you want to calculate the sum.
4. Click "Calculate Sequence": The calculator will instantly display the value of the nth term, the sum of the first n terms, and list the first few terms of the sequence.

Example:

Let's say you have an arithmetic sequence where the first term (a₁) is 2, and the common difference (d) is 3. You want to find the 10th term and the sum of the first 10 terms.

• First Term (a₁): 2
• Common Difference (d): 3
• Number of Terms (n): 10

Using the formulas:

• Nth Term (a₁₀):
  a₁₀ = 2 + (10 - 1) * 3
a₁₀ = 2 + 9 * 3
a₁₀ = 2 + 27 = 29
• Sum of First N Terms (S₁₀):
  S₁₀ = 10/2 * (2 * 2 + (10 - 1) * 3)
S₁₀ = 5 * (4 + 9 * 3)
S₁₀ = 5 * (4 + 27)
S₁₀ = 5 * 31 = 155

The calculator would show the 10th term as 29.00 and the sum of the first 10 terms as 155.00. The sequence would start: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.