1. What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. It's widely used in mathematics, finance, physics, and computer science.

2. How Does the Calculator Work?

The calculator uses the geometric sequence formula:

Where:

• \( a_n \) — The nth term of the sequence
• \( a \) — The first term of the sequence
• \( r \) — The common ratio between terms
• \( n \) — The term number (positive integer)

Explanation: The formula calculates any term in a geometric sequence by starting with the first term and multiplying by the common ratio raised to the power of (term number - 1).

3. Importance of Geometric Sequences

Details: Geometric sequences model exponential growth or decay patterns found in population growth, radioactive decay, interest calculations, and computer algorithms. Understanding them is fundamental in many STEM fields.

4. Using the Calculator

Tips: Enter the first term (a), common ratio (r), and term number (n). The term number must be a positive integer. All values can be positive or negative (except term number which must be positive).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between arithmetic and geometric sequences?
A: Arithmetic sequences add a constant difference each term, while geometric sequences multiply by a constant ratio.

Q2: Can the common ratio be negative?
A: Yes, negative ratios create alternating sequences (e.g., 1, -2, 4, -8, ...).

Q3: What happens when the common ratio is between 0 and 1?
A: The sequence shows exponential decay, with terms getting smaller in absolute value.

Q4: How is this used in real life?
A: Applications include calculating compound interest, modeling population growth, and analyzing algorithmic complexity.

Q5: What's the sum of the first n terms?
A: The sum S_n = a(1 - rⁿ)/(1 - r) when r ≠ 1, or S_n = n×a when r = 1.