Geometric Sequence Calculator

Geometric Sequence Calculator: Explanation and Guide

The Geometric Sequence Calculator is a powerful tool designed to compute terms, common ratio, finite sums, and infinite sums of a geometric sequence based on the inputs provided. It simplifies the process of solving geometric sequence-related problems, delivering step-by-step solutions for better understanding.

What Is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called 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: Compute specific terms of the geometric sequence.
• Find the Common Ratio: Determine 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)), compute the infinite sum ((S_\infty)).
• Step-by-Step Solutions: Get a detailed explanation for each calculation.

How to Use the Calculator

1. Input Data:
2. Enter the formula for (a_n) or provide the first three terms of the sequence.
3. Specify the common ratio ((r)) if known.

4. Optional: Enter the number of terms ((n)) for which you want the sum.

5. Examples Dropdown:

6. Use the Examples dropdown to select predefined data to see how the calculator works.

7. Calculate:

8. Click the Calculate button to compute the results.

9. Results will include terms, the common ratio, the sum of (n) terms, and the infinite sum (if it exists).

10. Clear Inputs:

11. Click Clear to reset all inputs and outputs.

Outputs

The calculator provides: - Terms: Displays the terms of the sequence based on the inputs. - Common Ratio: Shows 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: Provides detailed calculations for transparency 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 obtained 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 works.

This tool is ideal for students, educators, and anyone looking to simplify geometric sequence calculations. Let the Geometric Sequence Calculator do the math for you!