Arithmetic Sequence Calculator

Arithmetic Sequence Calculator

Calculation Results

The n-th term (aₙ):

The sum of n terms (Sₙ):

Sequence Preview (First 5 terms):

function calculateSequence() { var a1 = parseFloat(document.getElementById('firstTerm').value); var d = parseFloat(document.getElementById('commonDifference').value); var n = parseInt(document.getElementById('numTerms').value); if (isNaN(a1) || isNaN(d) || isNaN(n)) { alert("Please enter valid numeric values for all fields."); return; } if (n <= 0) { alert("The number of terms must be greater than zero."); return; } // Calculate n-th term: an = a1 + (n – 1) * d var an = a1 + (n – 1) * d; // Calculate sum: Sn = (n / 2) * (a1 + an) var sn = (n / 2) * (a1 + an); // Generate sequence preview var preview = []; var maxPreview = Math.min(n, 5); for (var i = 0; i 5) { previewString += ', … ' + an; } // Display results document.getElementById('nthTermVal').innerText = an.toLocaleString(); document.getElementById('sumTermsVal').innerText = sn.toLocaleString(); document.getElementById('sequencePreview').innerText = previewString; document.getElementById('resultArea').style.display = 'block'; }

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is a constant. This constant value is known as the common difference (d). Arithmetic sequences are fundamental in mathematics, appearing in everything from simple counting to complex physics equations and financial projections.

The Key Formulas

To calculate any property of an arithmetic sequence, you need three primary pieces of information: the first term (a₁), the common difference (d), and the position of the term you are looking for (n).

• N-th Term Formula: aₙ = a₁ + (n - 1)d
  Use this to find any specific number in the sequence without listing all the previous numbers.
• Sum of N Terms Formula: Sₙ = (n/2)(a₁ + aₙ)
  Also known as the arithmetic series formula, this calculates the total sum of all terms up to the n-th position.

Example Calculation

Let's say you have a sequence that starts with 10 and adds 5 every time. You want to find the 20th term and the sum of the first 20 terms.

1. Identify variables: a₁ = 10, d = 5, n = 20.
2. Find the 20th term (a₂₀): 10 + (20 – 1) * 5 = 10 + 95 = 105.
3. Find the sum (S₂₀): (20 / 2) * (10 + 105) = 10 * 115 = 1,150.

Common Applications

Arithmetic sequences aren't just for textbooks. They are used in real-world scenarios such as:

• Simple Interest: When interest is earned only on the principal amount, the total balance grows in an arithmetic sequence.
• Uniform Motion: If an object travels at a constant velocity, the distance covered in each second forms an arithmetic progression.
• Standardized Increases: Calculating stadium seating where each row has a set number of additional seats compared to the one in front.
• Salary Steps: Many professional contracts offer fixed annual raises, which follow an arithmetic sequence.