Sequence Calculator

Sequence Calculator

function updateInputVisibility() { var arithmeticInputs = document.querySelectorAll('.arithmetic-input'); var geometricInputs = document.querySelectorAll('.geometric-input'); var arithmeticRadio = document.getElementById('arithmetic'); if (arithmeticRadio.checked) { arithmeticInputs.forEach(function(el) { el.style.display = 'block'; }); geometricInputs.forEach(function(el) { el.style.display = 'none'; }); } else { arithmeticInputs.forEach(function(el) { el.style.display = 'none'; }); geometricInputs.forEach(function(el) { el.style.display = 'block'; }); } } document.addEventListener('DOMContentLoaded', updateInputVisibility); document.getElementById('arithmetic').addEventListener('change', updateInputVisibility); document.getElementById('geometric').addEventListener('change', updateInputVisibility); function calculateSequence() { var firstTermInput = document.getElementById('firstTerm').value; var numTermsInput = document.getElementById('numTerms').value; var commonDifferenceInput = document.getElementById('commonDifference').value; var commonRatioInput = document.getElementById('commonRatio').value; var a1 = parseFloat(firstTermInput); var n = parseInt(numTermsInput); var d = parseFloat(commonDifferenceInput); var r = parseFloat(commonRatioInput); var nthTermResult = document.getElementById('nthTermResult'); var sumNTermsResult = document.getElementById('sumNTermsResult'); var sequenceListResult = document.getElementById('sequenceListResult'); nthTermResult.textContent = '-'; sumNTermsResult.textContent = '-'; sequenceListResult.innerHTML = "; if (isNaN(a1) || isNaN(n) || n <= 0) { alert('Please enter valid numbers for First Term and a positive integer for Number of Terms.'); return; } var sequenceType = document.querySelector('input[name="sequenceType"]:checked').value; var an, Sn; var terms = []; if (sequenceType === 'arithmetic') { if (isNaN(d)) { alert('Please enter a valid number for Common Difference.'); return; } // Calculate nth term for arithmetic sequence: an = a1 + (n – 1)d an = a1 + (n – 1) * d; // Calculate sum of n terms for arithmetic sequence: Sn = n/2 * (2a1 + (n – 1)d) Sn = (n / 2) * (2 * a1 + (n – 1) * d); // Generate terms for (var i = 0; i < n; i++) { terms.push(a1 + i * d); } } else if (sequenceType === 'geometric') { if (isNaN(r)) { alert('Please enter a valid number for Common Ratio.'); return; } // Calculate nth term for geometric sequence: an = a1 * r^(n-1) an = a1 * Math.pow(r, n – 1); // Calculate sum of n terms for geometric sequence: Sn = a1 * (1 – r^n) / (1 – r) if (r === 1) { Sn = n * a1; // Special case for r = 1 } else { Sn = a1 * (1 – Math.pow(r, n)) / (1 – r); } // Generate terms for (var i = 0; i < n; i++) { terms.push(a1 * Math.pow(r, i)); } } nthTermResult.textContent = an.toFixed(4); sumNTermsResult.textContent = Sn.toFixed(4); for (var j = 0; j < terms.length; j++) { var listItem = document.createElement('li'); listItem.textContent = terms[j].toFixed(4); sequenceListResult.appendChild(listItem); } }

Understanding Sequences: Arithmetic and Geometric Progressions

A sequence is an ordered list of numbers, often defined by a specific rule or pattern. Sequences are fundamental concepts in mathematics and appear in various fields, from finance to computer science. This calculator helps you explore two of the most common types: Arithmetic Sequences and Geometric Sequences.

What is an Arithmetic Sequence?

An arithmetic sequence (also known as an arithmetic progression) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Key Formulas for Arithmetic Sequences:

• Nth Term (a_n): The formula to find any term in an arithmetic sequence is:
  an = a₁ + (n - 1)d
  Where:
  • a₁ is the first term
  • n is the term number you want to find
  • d is the common difference
• Sum of N Terms (S_n): The sum of the first n terms of an arithmetic sequence is:
  Sn = n/2 * (2a₁ + (n - 1)d)
  Alternatively, if you know the first and nth terms:
  Sn = n/2 * (a₁ + an)

Example: Consider an arithmetic sequence with a first term (a₁) of 3 and a common difference (d) of 4.

• The sequence would be: 3, 7, 11, 15, 19, …
• To find the 5th term (a₅): a₅ = 3 + (5 - 1) * 4 = 3 + 4 * 4 = 3 + 16 = 19
• To find the sum of the first 5 terms (S₅): S₅ = 5/2 * (2*3 + (5 - 1)*4) = 2.5 * (6 + 16) = 2.5 * 22 = 55

What is a Geometric Sequence?

A geometric sequence (also known as a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

Key Formulas for Geometric Sequences:

• Nth Term (a_n): The formula to find any term in a geometric sequence is:
  an = a₁ * r^(n-1)
  Where:
  • a₁ is the first term
  • n is the term number you want to find
  • r is the common ratio
• Sum of N Terms (S_n): The sum of the first n terms of a geometric sequence is:
  Sn = a₁ * (1 - r^n) / (1 - r) (where r ≠ 1)
  If r = 1, then Sn = n * a₁

Example: Consider a geometric sequence with a first term (a₁) of 2 and a common ratio (r) of 3.

• The sequence would be: 2, 6, 18, 54, 162, …
• To find the 5th term (a₅): a₅ = 2 * 3^(5-1) = 2 * 3^4 = 2 * 81 = 162
• To find the sum of the first 5 terms (S₅): S₅ = 2 * (1 - 3^5) / (1 - 3) = 2 * (1 - 243) / (-2) = 2 * (-242) / (-2) = 242

How to Use the Sequence Calculator

1. Select Sequence Type: Choose whether you want to calculate an "Arithmetic Sequence" or a "Geometric Sequence" using the radio buttons.
2. Enter First Term (a₁): Input the starting value of your sequence.
3. Enter Number of Terms (n): Specify how many terms you want to calculate and sum.
4. Enter Common Difference (d) or Common Ratio (r):
   • If "Arithmetic Sequence" is selected, enter the constant difference between terms.
   • If "Geometric Sequence" is selected, enter the constant multiplier between terms.
5. Click "Calculate Sequence": The calculator will instantly display the Nth term, the sum of the first N terms, and a list of all the terms in the sequence.

This tool is perfect for students, educators, or anyone needing quick calculations for sequences in mathematics, finance, or other analytical tasks.