Geometric Sequence Calculator

Geometric Sequence Calculator

function calculateGeometricSequence() { var firstTerm = parseFloat(document.getElementById('firstTerm').value); var commonRatio = parseFloat(document.getElementById('commonRatio').value); var numTerms = parseInt(document.getElementById('numTerms').value); if (isNaN(firstTerm) || isNaN(commonRatio) || isNaN(numTerms) || numTerms < 1) { document.getElementById('nthTermResult').textContent = 'Please enter valid numbers for all fields.'; document.getElementById('sumTermsResult').textContent = ''; document.getElementById('sequenceListResult').textContent = ''; return; } var nthTerm; var sumTerms; var sequence = []; // Calculate nth term nthTerm = firstTerm * Math.pow(commonRatio, numTerms – 1); // Calculate sum of first n terms if (commonRatio === 1) { sumTerms = firstTerm * numTerms; } else { sumTerms = firstTerm * (1 – Math.pow(commonRatio, numTerms)) / (1 – commonRatio); } // Generate sequence list for (var i = 0; i < numTerms; i++) { sequence.push(firstTerm * Math.pow(commonRatio, i)); } document.getElementById('nthTermResult').textContent = nthTerm.toFixed(4); document.getElementById('sumTermsResult').textContent = sumTerms.toFixed(4); document.getElementById('sequenceListResult').textContent = sequence.map(function(term) { return term.toFixed(4); }).join(', '); }

Understanding Geometric Sequences

A geometric sequence, also known as a geometric progression, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence is fundamental in various fields, from mathematics and physics to finance and computer science.

Key Components of a Geometric Sequence

• First Term (a): This is the starting number of the sequence. It's the value from which all subsequent terms are derived.
• Common Ratio (r): This is the constant factor by which each term is multiplied to get the next term. The common ratio can be positive or negative, an integer or a fraction.
• Number of Terms (n): This indicates how many terms are in the sequence you are considering.

Formulas for Geometric Sequences

There are two primary formulas used to analyze geometric sequences:

1. The Nth Term Formula

To find any specific term (the n-th term) in a geometric sequence, you use the formula:

a_n = a * r^(n-1)

Where:

• an is the n-th term you want to find.
• a is the first term.
• r is the common ratio.
• n is the term number (e.g., 1st, 2nd, 3rd, etc.).

2. The Sum of the First N Terms Formula

To find the sum of the first 'n' terms of a geometric sequence, there are two cases:

Case 1: When the common ratio (r) is not equal to 1

S_n = a * (1 – rⁿ) / (1 – r)

Case 2: When the common ratio (r) is equal to 1

S_n = a * n

Where:

• Sn is the sum of the first n terms.
• a is the first term.
• r is the common ratio.
• n is the number of terms.

How to Use the Geometric Sequence Calculator

Our calculator simplifies the process of working with geometric sequences:

1. Enter the First Term (a): Input the starting value of your sequence.
2. Enter the Common Ratio (r): Input the number by which each term is multiplied to get the next.
3. Enter the Number of Terms (n): Specify how many terms you want to consider in the sequence.
4. Click "Calculate Sequence": The calculator will instantly display the n-th term, the sum of the first n terms, and a list of all terms in the sequence.

Examples of Geometric Sequences

Example 1: Simple Growth

Consider a sequence starting with 2, and each subsequent term is multiplied by 3.

• First Term (a) = 2
• Common Ratio (r) = 3
• Number of Terms (n) = 5

Using the calculator, you would find:

• Nth Term (a₅) = 2 * 3^(5-1) = 2 * 3⁴ = 2 * 81 = 162
• Sum of First N Terms (S₅) = 2 * (1 – 3⁵) / (1 – 3) = 2 * (1 – 243) / (-2) = 2 * (-242) / (-2) = 242
• Sequence List: 2, 6, 18, 54, 162

Example 2: Decreasing Sequence

Let's look at a sequence where terms decrease.

• First Term (a) = 100
• Common Ratio (r) = 0.5
• Number of Terms (n) = 4

The calculator would show:

• Nth Term (a₄) = 100 * 0.5^(4-1) = 100 * 0.5³ = 100 * 0.125 = 12.5
• Sum of First N Terms (S₄) = 100 * (1 – 0.5⁴) / (1 – 0.5) = 100 * (1 – 0.0625) / 0.5 = 100 * 0.9375 / 0.5 = 187.5
• Sequence List: 100, 50, 25, 12.5

Example 3: Common Ratio of 1

What happens if the common ratio is 1?

• First Term (a) = 7
• Common Ratio (r) = 1
• Number of Terms (n) = 3

The calculator would yield:

• Nth Term (a₃) = 7 * 1^(3-1) = 7 * 1 = 7
• Sum of First N Terms (S₃) = 7 * 3 = 21 (using the special case formula)
• Sequence List: 7, 7, 7

This calculator is a powerful tool for students, educators, and professionals needing to quickly analyze and understand geometric sequences without manual calculations.