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 fixed, non-zero number called the common ratio. It follows the pattern: a, ar, ar², ar³, ...

2. How Does the Calculator Work?

The calculator uses the geometric sequence formula:

Where:

• \( a_1 \) — First term of the sequence
• \( r \) — Common ratio between terms
• \( n \) — Position of the term in the sequence
• \( a_n \) — The nth term of the sequence

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

3. Importance of Geometric Sequences

Details: Geometric sequences are fundamental in mathematics and have applications in finance (compound interest), computer science (algorithm analysis), physics (exponential decay), and many other fields.

4. Using the Calculator

Tips: Enter the first term (a₁), common ratio (r), and the term position (n) you want to calculate. All values are unitless. The term position must be a positive integer.

5. Frequently Asked Questions (FAQ)

Q1: What if the common ratio is negative?
A: The sequence will alternate between positive and negative values, but the formula still applies correctly.

Q2: Can the common ratio be zero?
A: Yes, but if r = 0, all terms after the first will be zero.

Q3: What if the common ratio is between 0 and 1?
A: The sequence will be decreasing (if a₁ > 0) and will approach zero as n increases.

Q4: How is this different from an arithmetic sequence?
A: Arithmetic sequences add a constant difference, while geometric sequences multiply by a constant ratio.

Q5: Can I calculate the sum of geometric sequence terms?
A: Yes, but this calculator only finds individual terms. The sum formula is different: Sₙ = a₁(1-rⁿ)/(1-r) for r ≠ 1.