The key difference between a finite and infinite geometric sequence lies in the number of terms.
- Finite geometric sequence: This sequence has a limited number of terms. It has a defined beginning and end.
- Infinite geometric sequence: This sequence has an unlimited number of terms. It continues indefinitely.
Understanding the concepts:
- Geometric sequence: A sequence where each term is found by multiplying the previous term by a constant value, known as the common ratio.
- Finite: Having a definite end.
- Infinite: Having no end.
Examples:
- Finite geometric sequence: 2, 4, 8, 16, 32 (common ratio = 2, 5 terms)
- Infinite geometric sequence: 1, 1/2, 1/4, 1/8, ... (common ratio = 1/2, infinite terms)
Practical Insights:
- Finite geometric sequences are often used in situations where there is a clear starting and ending point, like the amount of money in a bank account after a certain number of years with compound interest.
- Infinite geometric sequences can model situations where there is no defined end, like the decay of radioactive material or the spread of a virus.
Solutions:
- Determining if a sequence is finite or infinite: Analyze the pattern and the number of terms given.
- Calculating the sum of a finite geometric sequence: Use the formula: Sn = a(1-r^n)/(1-r), where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.
- Determining the convergence of an infinite geometric sequence: An infinite geometric sequence converges if the absolute value of the common ratio is less than 1 (|r| < 1). The sum of an infinite convergent geometric sequence is given by: S = a/(1-r).
