Find Explicit Formula Calculator

Find the Explicit Formula (a_n)

Enter the details of your sequence to find its explicit formula and see the first few terms.

Results copied to clipboard!

What is an Explicit Formula Calculator?

An Explicit Formula Calculator is a tool designed to determine the explicit formula (often denoted as a_n) for a given sequence, typically either arithmetic or geometric. An explicit formula allows you to find the value of any term in a sequence directly, just by knowing its position (n), without needing to calculate all preceding terms. This is in contrast to a recursive formula, which defines a term based on the previous term(s).

This calculator is particularly useful for students learning about sequences, mathematicians, engineers, and anyone dealing with patterns that can be described by arithmetic or geometric progressions. The Explicit Formula Calculator helps you visualize and understand the behavior of these sequences.

Common misconceptions include thinking that all sequences have a simple explicit formula (some are more complex or don't have one) or confusing explicit formulas with recursive ones. Our Explicit Formula Calculator focuses on the two most common types: arithmetic and geometric.

Explicit Formula and Mathematical Explanation

The explicit formula depends on whether the sequence is arithmetic or geometric.

Arithmetic Sequence

In an arithmetic sequence, each term after the first is obtained by adding a constant difference, 'd', to the preceding term. The explicit formula for an arithmetic sequence is:

a_n = a₁ + (n – 1)d

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number
• d is the common difference

Geometric Sequence

In a geometric sequence, each term after the first is obtained by multiplying the preceding term by a constant non-zero ratio, 'r'. The explicit formula for a geometric sequence is:

a_n = a₁ * r^{(n – 1)}

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number
• r is the common ratio

Variables Table

Variable	Meaning	Unit	Typical Range
an	The value of the nth term in the sequence	Varies (unitless or based on a1)	Varies
a1	The first term of the sequence	Varies	Any real number
n	The term number or position in the sequence	Integer	Positive integers (1, 2, 3, …)
d	The common difference (for arithmetic)	Varies (same as a1)	Any real number
r	The common ratio (for geometric)	Varies (unitless if a1 is unitless)	Any real number except 0 (often |r| != 1)

Our Explicit Formula Calculator uses these formulas to derive the explicit rule for your sequence.

Practical Examples (Real-World Use Cases)

Let's see how the Explicit Formula Calculator works with some examples.

Example 1: Arithmetic Sequence

Suppose you are saving money. You start with $100 (a₁ = 100) and decide to add $20 each month (d = 20). You want to find the formula for how much money you have after 'n' months.

• Type of Sequence: Arithmetic
• First Term (a₁): 100
• Common Difference (d): 20

Using the Explicit Formula Calculator (or the formula a_n = a₁ + (n – 1)d), we get:

a_n = 100 + (n – 1)20 = 100 + 20n – 20 = 80 + 20n

So, the explicit formula is a_n = 80 + 20n (or a_n = 20n + 80). After 12 months (n=12), you would have a₁₂ = 80 + 20*12 = 80 + 240 = $320.

Example 2: Geometric Sequence

Imagine a population of bacteria that doubles every hour. You start with 50 bacteria (a₁ = 50), and the population doubles (r = 2) each hour.

• Type of Sequence: Geometric
• First Term (a₁): 50
• Common Ratio (r): 2

Using the Explicit Formula Calculator (or the formula a_n = a₁ * r^{(n – 1)}), we get:

a_n = 50 * 2^{(n – 1)}

After 5 hours (n=5), the population would be a₅ = 50 * 2^{(5 – 1)} = 50 * 2⁴ = 50 * 16 = 800 bacteria. Our sequence calculator can quickly show this.

How to Use This Explicit Formula Calculator

Using our Explicit Formula Calculator is straightforward:

1. Select Sequence Type: Choose whether you are working with an "Arithmetic" or "Geometric" sequence from the dropdown menu. The appropriate input fields for common difference or ratio will appear.
2. Enter First Term (a₁): Input the very first number in your sequence.
3. Enter Common Difference (d) or Ratio (r): If you selected "Arithmetic," enter the common difference 'd'. If you selected "Geometric," enter the common ratio 'r' (it cannot be zero).
4. Enter Number of Terms (n): Specify how many terms of the sequence you want to see displayed in the results table and chart (between 2 and 50).
5. Calculate: The calculator automatically updates as you type, or you can click the "Calculate" button.
6. View Results: The calculator will display:
   • The derived explicit formula for a_n.
   • The input values (a₁, d or r, type).
   • A table showing the first 'n' terms of the sequence.
   • A chart visualizing these terms.
7. Reset: Click "Reset" to clear the inputs to default values.
8. Copy Results: Click "Copy Results" to copy the formula, inputs, and the first few terms to your clipboard.

The results help you understand the pattern and predict future terms. The sequence chart provides a visual representation.

Key Factors That Affect Explicit Formula Results

Several factors influence the explicit formula and the terms of a sequence calculated by the Explicit Formula Calculator:

1. Type of Sequence: Whether it's arithmetic or geometric fundamentally changes the formula (linear vs. exponential growth/decay).
2. First Term (a₁): This is the starting point of the sequence and directly scales or shifts all subsequent terms.
3. Common Difference (d): In arithmetic sequences, a larger 'd' means faster linear growth or decay. A positive 'd' means increasing terms, negative 'd' means decreasing.
4. Common Ratio (r): In geometric sequences:
   • If |r| > 1, the sequence grows exponentially (diverges).
   • If |r| < 1 (and r != 0), the sequence decays exponentially towards zero (converges).
   • If r = 1, it's a constant sequence.
   • If r < 0, the terms alternate in sign.
   • r cannot be 0 for a standard geometric sequence definition handled here.
5. The Term Number (n): As 'n' increases, the value of a_n changes according to the formula, moving further from a₁ based on 'd' or 'r'.
6. Initial Conditions: The starting term (a₁) and the nature of progression (d or r) are the critical initial conditions defining the entire sequence. Understanding the explicit rule for sequence is key.

The Explicit Formula Calculator clearly shows how these factors combine.

Frequently Asked Questions (FAQ)

What is the difference between an explicit and a recursive formula?

An explicit formula allows you to calculate any term a_n directly using 'n' (e.g., a_n = 2n + 1). A recursive formula defines a term based on the preceding term(s) (e.g., a_n = a_n-1 + 2, with a₁ given). Our Explicit Formula Calculator focuses on finding the explicit form.

Can I use the calculator for sequences that are neither arithmetic nor geometric?

No, this Explicit Formula Calculator is specifically designed for arithmetic and geometric sequences. Other types of sequences (like quadratic or Fibonacci) require different methods to find their explicit formulas.

What if my common ratio 'r' is 0 or 1?

If 'r' is 1, the geometric sequence becomes a constant sequence (a₁, a₁, a₁,…). If 'r' is 0, after the first term, all subsequent terms become 0, which is a trivial case usually excluded from the definition where 'r' is non-zero. Our calculator handles r=1 but flags r=0 as invalid for geometric sequences.

How do I find the common difference or ratio if I only have a few terms?

For an arithmetic sequence, subtract any term from its succeeding term (d = a_n – a_n-1). For a geometric sequence, divide any term by its preceding term (r = a_n / a_n-1), provided a_n-1 is not zero. You need to check if this difference or ratio is constant across several pairs of terms.

Can the first term a₁ be zero?

Yes, the first term can be zero for both arithmetic and geometric sequences. If a₁=0 in a geometric sequence, all terms will be zero unless r is undefined (division by zero).

What does the chart represent?

The chart plots the term number 'n' on the x-axis and the corresponding term value 'a_n' on the y-axis, giving a visual representation of how the sequence grows or decays. It helps visualize the find nth term concept.

Can I find the sum of the first 'n' terms with this calculator?

No, this Explicit Formula Calculator focuses on finding the formula for a_n and displaying terms. To find the sum, you would need formulas for the sum of the first 'n' terms of an arithmetic or geometric series (S_n).

How accurate is the Explicit Formula Calculator?

The calculator is accurate based on the standard formulas for arithmetic and geometric sequences, using standard floating-point arithmetic. For very large 'n' or extreme values of 'r', numerical precision limits might be encountered, but for typical use, it's very accurate.