/* Basic styling for the calculator */
.arithmetic-sequence-calculator {
font-family: Arial, sans-serif;
max-width: 600px;
margin: 20px auto;
padding: 20px;
border: 1px solid #ccc;
border-radius: 8px;
background-color: #f9f9f9;
}
.arithmetic-sequence-calculator h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
}
.arithmetic-sequence-calculator label {
display: block;
margin-bottom: 8px;
font-weight: bold;
color: #555;
}
.arithmetic-sequence-calculator input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
margin-bottom: 15px;
border: 1px solid #ddd;
border-radius: 4px;
box-sizing: border-box;
}
.arithmetic-sequence-calculator button {
display: block;
width: 100%;
padding: 12px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 16px;
cursor: pointer;
transition: background-color 0.3s ease;
}
.arithmetic-sequence-calculator button:hover {
background-color: #0056b3;
}
.arithmetic-sequence-calculator .results {
margin-top: 25px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 4px;
background-color: #eaf4ff;
}
.arithmetic-sequence-calculator .results p {
margin-bottom: 8px;
line-height: 1.6;
color: #333;
}
.arithmetic-sequence-calculator .results p strong {
color: #0056b3;
}
.arithmetic-sequence-calculator .error {
color: red;
margin-top: 10px;
font-weight: bold;
}
.arithmetic-sequence-calculator .sequence-display {
margin-top: 15px;
padding: 10px;
background-color: #f0f8ff;
border: 1px dashed #a0c0e0;
border-radius: 4px;
font-family: 'Courier New', monospace;
word-wrap: break-word;
}
function calculateArithmeticSequence() {
var firstTermInput = document.getElementById("firstTerm").value;
var commonDifferenceInput = document.getElementById("commonDifference").value;
var numberOfTermsInput = document.getElementById("numberOfTerms").value;
var errorMessageDiv = document.getElementById("errorMessage");
errorMessageDiv.textContent = ""; // Clear previous errors
var a1 = parseFloat(firstTermInput);
var d = parseFloat(commonDifferenceInput);
var n = parseInt(numberOfTermsInput);
if (isNaN(a1) || isNaN(d) || isNaN(n)) {
errorMessageDiv.textContent = "Please enter valid numbers for all fields.";
document.getElementById("nthTermResult").textContent = "";
document.getElementById("sumOfNTermsResult").textContent = "";
document.getElementById("sequenceResult").textContent = "";
return;
}
if (n < 1) {
errorMessageDiv.textContent = "Number of Terms (n) must be at least 1.";
document.getElementById("nthTermResult").textContent = "";
document.getElementById("sumOfNTermsResult").textContent = "";
document.getElementById("sequenceResult").textContent = "";
return;
}
// Calculate Nth Term (a_n = a_1 + (n – 1)d)
var an = a1 + (n – 1) * d;
// Calculate Sum of N Terms (S_n = n/2 * (a_1 + a_n))
var Sn = (n / 2) * (a1 + an);
document.getElementById("nthTermResult").textContent = an.toFixed(4); // Display with 4 decimal places
document.getElementById("sumOfNTermsResult").textContent = Sn.toFixed(4); // Display with 4 decimal places
// Generate and display the sequence
var sequence = [];
var termsToShowStart = 4; // How many terms to show at the beginning
var termsToShowEnd = 1; // How many terms to show at the end (just the last one)
if (n === 1) {
sequence.push(a1.toFixed(4));
} else if (n <= termsToShowStart + termsToShowEnd) { // If n is small enough to show all terms
for (var i = 0; i < n; i++) {
sequence.push((a1 + i * d).toFixed(4));
}
} else { // If n is large, show start, …, end
for (var i = 0; i < termsToShowStart; i++) {
sequence.push((a1 + i * d).toFixed(4));
}
sequence.push("…");
sequence.push(an.toFixed(4)); // The nth term
}
document.getElementById("sequenceResult").textContent = "Sequence: " + sequence.join(", ");
}
Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'. Each term after the first is obtained by adding the common difference to the preceding term.
Key Components of an Arithmetic Sequence:
- First Term (a₁): The starting number of the sequence.
- Common Difference (d): The constant value added to each term to get the next term.
- Number of Terms (n): The position of a specific term in the sequence, or the total count of terms being considered.
- Nth Term (aₙ): The value of the term at the 'n-th' position in the sequence.
- Sum of N Terms (Sₙ): The total sum of the first 'n' terms of the sequence.
Formulas for Arithmetic Sequences:
The two primary formulas used to work with arithmetic sequences are:
- Formula for the Nth Term (aₙ):
aₙ = a₁ + (n - 1)d
This formula allows you to find any term in the sequence, given the first term, the common difference, and the term's position.
- Formula for the Sum of N Terms (Sₙ):
Sₙ = n/2 * (a₁ + aₙ)
Alternatively, if you don't know aₙ, you can use:
Sₙ = n/2 * (2a₁ + (n - 1)d)
This formula calculates the sum of all terms from the first term up to the n-th term.
How to Use the Calculator:
Our Arithmetic Sequence Calculator simplifies finding the n-th term and the sum of the first n terms. Simply input the following values:
- First Term (a₁): Enter the initial value of your sequence.
- Common Difference (d): Input the constant difference between consecutive terms.
- Number of Terms (n): Specify which term you want to find (e.g., 5 for the 5th term) and how many terms you want to sum.
Click "Calculate," and the calculator will instantly provide the value of the n-th term, the sum of the first n terms, and display the beginning of the sequence.
Example:
Let's consider an arithmetic sequence where the first term is 2, and the common difference is 3. We want to find the 5th term and the sum of the first 5 terms.
- First Term (a₁): 2
- Common Difference (d): 3
- Number of Terms (n): 5
Using the formulas:
- Nth Term (a₅):
a₅ = a₁ + (5 - 1)d
a₅ = 2 + (4) * 3
a₅ = 2 + 12
a₅ = 14
- Sum of N Terms (S₅):
First, find a₅ (which we did above, a₅ = 14).
S₅ = 5/2 * (a₁ + a₅)
S₅ = 2.5 * (2 + 14)
S₅ = 2.5 * 16
S₅ = 40
The sequence would be: 2, 5, 8, 11, 14.
Using the calculator with these inputs will yield the same results, making complex calculations quick and easy.