Calculator for Sequences

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.seq-calculator-container { font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; max-width: 800px; margin: 20px auto; padding: 25px; border: 1px solid #e0e0e0; border-radius: 12px; background-color: #ffffff; box-shadow: 0 4px 6px rgba(0,0,0,0.05); } .seq-calculator-container h2 { color: #2c3e50; margin-top: 0; text-align: center; font-size: 24px; } .seq-input-group { margin-bottom: 20px; } .seq-input-group label { display: block; margin-bottom: 8px; font-weight: 600; color: #34495e; } .seq-input-group select, .seq-input-group input { width: 100%; padding: 12px; border: 2px solid #ddd; border-radius: 6px; font-size: 16px; box-sizing: border-box; transition: border-color 0.3s; } .seq-input-group input:focus { border-color: #3498db; outline: none; } .seq-btn { width: 100%; background-color: #3498db; color: white; padding: 14px; border: none; border-radius: 6px; font-size: 18px; font-weight: bold; cursor: pointer; transition: background-color 0.3s; } .seq-btn:hover { background-color: #2980b9; } .seq-result-box { margin-top: 25px; padding: 20px; background-color: #f8f9fa; border-left: 5px solid #3498db; border-radius: 4px; } .seq-result-item { margin-bottom: 10px; font-size: 16px; } .seq-result-item span { font-weight: bold; color: #2c3e50; } .seq-article { margin-top: 40px; line-height: 1.6; color: #444; } .seq-article h3 { color: #2c3e50; border-bottom: 2px solid #eee; padding-bottom: 10px; } .seq-article table { width: 100%; border-collapse: collapse; margin: 20px 0; } .seq-article th, .seq-article td { border: 1px solid #ddd; padding: 12px; text-align: left; } .seq-article th { background-color: #f2f2f2; }

Sequence and Series Calculator

Arithmetic (Addition/Subtraction) Geometric (Multiplication/Division)
Sequence Type:
The n-th term (aₙ):
Sum of first n terms (Sₙ):
Sequence Preview:
document.getElementById('sequenceType').onchange = function() { var label = document.getElementById('diffRatioLabel'); var input = document.getElementById('diffRatio'); if (this.value === 'arithmetic') { label.innerText = 'Common Difference (d)'; input.placeholder = 'e.g., 3'; } else { label.innerText = 'Common Ratio (r)'; input.placeholder = 'e.g., 2'; } }; function calculateSeq() { var type = document.getElementById('sequenceType').value; var a1 = parseFloat(document.getElementById('firstTerm').value); var dr = parseFloat(document.getElementById('diffRatio').value); var n = parseInt(document.getElementById('numTerms').value); var resBox = document.getElementById('seqResult'); if (isNaN(a1) || isNaN(dr) || isNaN(n) || n <= 0) { alert('Please enter valid numerical values. Number of terms must be greater than 0.'); return; } var nthTerm = 0; var sumN = 0; var preview = []; var previewCount = Math.min(n, 5); if (type === 'arithmetic') { // a_n = a1 + (n-1)d nthTerm = a1 + (n – 1) * dr; // S_n = n/2 * (a1 + an) sumN = (n / 2) * (a1 + nthTerm); for (var i = 0; i < previewCount; i++) { preview.push((a1 + i * dr).toFixed(2).replace(/\.00$/, '')); } } else { // a_n = a1 * r^(n-1) nthTerm = a1 * Math.pow(dr, n – 1); // S_n = a1 * (1 – r^n) / (1 – r) if (dr === 1) { sumN = a1 * n; } else { sumN = a1 * (1 – Math.pow(dr, n)) / (1 – dr); } for (var j = 0; j 5 ? '…' : "); resBox.style.display = 'block'; }

Understanding Mathematical Sequences

A sequence is an ordered list of numbers that follows a specific pattern. Identifying the type of sequence is crucial for solving problems in algebra, calculus, and financial mathematics. The two most common types are Arithmetic and Geometric sequences.

Arithmetic Sequences

In an arithmetic sequence, the difference between any two consecutive terms is constant. This constant is called the Common Difference (d). If you know the first term and the difference, you can find any term in the series.

  • Formula for Nth Term: aₙ = a₁ + (n – 1)d
  • Formula for Sum: Sₙ = (n/2)(a₁ + aₙ)

Example: In the sequence 2, 5, 8, 11…, the first term (a₁) is 2 and the common difference (d) is 3.

Geometric Sequences

In a geometric sequence, each term is found by multiplying the previous term by a constant called the Common Ratio (r). These sequences grow or shrink exponentially.

  • Formula for Nth Term: aₙ = a₁ × r⁽ⁿ⁻¹⁾
  • Formula for Sum: Sₙ = a₁(1 – rⁿ) / (1 – r)

Example: In the sequence 3, 6, 12, 24…, the first term (a₁) is 3 and the common ratio (r) is 2.

Comparison Table

Feature Arithmetic Geometric
Relation between terms Addition/Subtraction Multiplication/Division
Growth Pattern Linear Exponential
Key Variable Difference (d) Ratio (r)

Practical Applications

Sequences are not just theoretical math. They are used in various fields:

  • Finance: Calculating simple interest (arithmetic) or compound interest and depreciation (geometric).
  • Computer Science: Analyzing algorithm complexity and memory allocation patterns.
  • Physics: Measuring radioactive decay or the bouncing height of a ball.
  • Biology: Modeling population growth over discrete time intervals.

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