Geometric Sequences Calculator

Geometric Sequences Calculator

function calculateGeometricSequence() { var firstTermInput = document.getElementById("firstTerm").value; var commonRatioInput = document.getElementById("commonRatio").value; var numTermsInput = document.getElementById("numTerms").value; var a1 = parseFloat(firstTermInput); var r = parseFloat(commonRatioInput); var n = parseInt(numTermsInput); var nthTermResultElement = document.getElementById("nthTermResult"); var sumTermsResultElement = document.getElementById("sumTermsResult"); if (isNaN(a1) || isNaN(r) || isNaN(n) || n <= 0) { nthTermResultElement.textContent = "Please enter valid numbers for all fields. Number of terms must be a positive integer."; sumTermsResultElement.textContent = ""; return; } // Calculate Nth Term (a_n = a_1 * r^(n-1)) var nthTerm = a1 * Math.pow(r, n – 1); nthTermResultElement.textContent = nthTerm.toFixed(4); // Display with 4 decimal places // Calculate Sum of First N Terms (S_n) var sumOfTerms; if (r === 1) { sumOfTerms = n * a1; } else { sumOfTerms = a1 * (1 – Math.pow(r, n)) / (1 – r); } sumTermsResultElement.textContent = sumOfTerms.toFixed(4); // Display with 4 decimal places } // Initial calculation on load with default values window.onload = calculateGeometricSequence;

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 (a₁): This is the starting number of the sequence.
• Common Ratio (r): This is the constant factor by which each term is multiplied to get the next term. It can be positive or negative, an integer or a fraction.
• Number of Terms (n): This indicates the position of a specific term in the sequence or the total number of terms being considered for a sum.

Formulas for Geometric Sequences

1. The Nth Term (a_n)

To find any specific term in a geometric sequence, you can use the formula:

a_n = a₁ * r^(n-1)

Where:

• a_n is the nth term you want to find.
• a₁ is the first term.
• r is the common ratio.
• n is the term number (its position in the sequence).

2. Sum of the First N Terms (S_n)

To find the sum of a certain number of terms in a geometric sequence, there are two formulas depending on the common ratio:

If the common ratio (r) is NOT equal to 1:

S_n = a₁ * (1 - r^n) / (1 - r)

If the common ratio (r) IS equal to 1:

S_n = n * a₁

Where:

• S_n is the sum of the first n terms.
• a₁ is the first term.
• r is the common ratio.
• n is the number of terms.

Example of a Geometric Sequence

Consider a geometric sequence starting with 2, where each subsequent term is three times the previous one. The sequence would look like: 2, 6, 18, 54, 162, …

• First Term (a₁): 2
• Common Ratio (r): 3

Let's use the calculator to find the 4th term and the sum of the first 4 terms:

• Input First Term (a₁): 2
• Input Common Ratio (r): 3
• Input Number of Terms (n): 4

Calculations:

• Nth Term (a₄): a₄ = 2 * 3^(4-1) = 2 * 3^3 = 2 * 27 = 54
• Sum of First N Terms (S₄): S₄ = 2 * (1 - 3^4) / (1 - 3) = 2 * (1 - 81) / (-2) = 2 * (-80) / (-2) = 160 / 2 = 80

The calculator above will quickly provide these results for any given inputs.

How to Use the Calculator

1. Enter the First Term (a₁): Input the starting value of your geometric sequence.
2. Enter the Common Ratio (r): Input the constant multiplier between consecutive terms.
3. Enter the Number of Terms (n): Specify which term you are interested in (for a_n) or how many terms you want to sum (for S_n).
4. Click "Calculate Sequence": The calculator will instantly display the value of the Nth term and the sum of the first N terms based on your inputs.

This tool simplifies complex calculations, allowing you to quickly analyze and understand geometric progressions without manual computation.