Geometric Sequence Calculator
Category: Sequences and SeriesCalculate terms, common ratio, sums, and infinite sums of a geometric sequence.
Geometric Sequence Calculator: Explanation and Guide
The Geometric Sequence Calculator is a handy tool designed to calculate terms, common ratio, finite sums, and infinite sums of a geometric sequence based on the inputs given. It makes solving problems related to geometric sequences easier, providing step-by-step solutions for clearer understanding.
What Is a Geometric Sequence?
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio ((r)).
For example: - Sequence: (2, 6, 18, 54) - Common ratio: (r = \frac{6}{2} = 3)
In general, the (n)-th term of a geometric sequence can be expressed as: [ a_n = a_1 \cdot r^{n-1} ] where: - (a_1) is the first term, - (r) is the common ratio, - (n) is the position of the term in the sequence.
Features of the Calculator
- Calculate Terms: Find specific terms of the geometric sequence.
- Find the Common Ratio: Identify the ratio between consecutive terms.
- Sum of (n) Terms: Calculate the sum of the first (n) terms ((S_n)).
- Infinite Sum: If applicable ((|r| < 1)), calculate the infinite sum ((S_\infty)).
- Step-by-Step Solutions: Receive a detailed explanation for each calculation.
How to Use the Calculator
- Input Data:
- Enter the formula for (a_n) or provide the first three terms of the sequence.
- Specify the common ratio ((r)) if you know it.
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Optional: Enter the number of terms ((n)) for which you want the sum.
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Examples Dropdown:
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Use the Examples dropdown to choose predefined data to see how the calculator operates.
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Calculate:
- Click the Calculate button to get the results.
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Results will include terms, the common ratio, the sum of (n) terms, and the infinite sum (if it exists).
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Clear Inputs:
- Click Clear to reset all inputs and outputs.
Outputs
The calculator provides: - Terms: Shows the terms of the sequence based on the inputs. - Common Ratio: Displays the fixed multiplier between terms. - Sum of (n) Terms ((S_n)): Calculates the sum using the formula: [ S_n = a_1 \frac{1 - r^n}{1 - r} \quad \text{(if (r \neq 1))} ] - Infinite Sum ((S_\infty)): Calculates the infinite sum for (|r| < 1) using: [ S_\infty = \frac{a_1}{1 - r} ] - Step-by-Step Explanation: Offers detailed calculations for clarity and learning.
Example Use Cases
Example 1
- Sequence: (2, 6, 18)
- Common Ratio: (r = 3)
- Sum of First 4 Terms: [ S_4 = 2 \frac{1 - 3^4}{1 - 3} = 80 ]
Example 2
- Formula: (a_n = 5 \cdot 2^{n-1})
- Sequence: (5, 10, 20, \dots)
- Infinite Sum: [ S_\infty = \frac{5}{1 - 2} \quad \text{(Not applicable since (|r| > 1))} ]
FAQ
What is a geometric sequence?
A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a fixed number, called the common ratio ((r)).
What is the common ratio?
The common ratio is the constant value by which each term in the sequence is multiplied to get the next term. It is calculated as: [ r = \frac{a_2}{a_1} ]
When does the infinite sum exist?
The infinite sum exists only when the absolute value of the common ratio is less than 1 ((|r| < 1)).
What is the sum of (n) terms ((S_n))?
The sum of the first (n) terms in a geometric sequence is calculated as: [ S_n = a_1 \frac{1 - r^n}{1 - r} \quad \text{if (r \neq 1)}. ]
What happens if the common ratio is 1?
If (r = 1), the sequence becomes constant, and the sum is: [ S_n = n \cdot a_1 ]
What does the dropdown do?
The dropdown provides pre-defined examples to help users understand how the calculator functions.
This tool is perfect for students, educators, and anyone looking to make geometric sequence calculations simpler. Let the Geometric Sequence Calculator handle the math for you!