Geometric Sequence

Definition

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 called the common ratio. This type of sequence has a predictable pattern that can be expressed mathematically, making it important for understanding limits and convergence in sequences.

5 Must Know Facts For Your Next Test

1. In a geometric sequence, if the common ratio is greater than 1, the terms will increase; if it's between 0 and 1, the terms will decrease towards zero.
2. The formula for the nth term of a geometric sequence can be expressed as $$a_n = a_1 imes r^{(n-1)}$$, where $$a_1$$ is the first term and $$r$$ is the common ratio.
3. The sum of an infinite geometric series can converge to a finite number only if the absolute value of the common ratio is less than 1.
4. Geometric sequences can model real-world situations such as population growth, interest calculations, and radioactive decay.
5. In terms of limits, the behavior of a geometric sequence can often be analyzed by looking at its common ratio and determining if it leads to convergence or divergence.

Review Questions

• How does the common ratio affect the behavior of a geometric sequence?
  • The common ratio significantly influences whether a geometric sequence will grow or shrink. If the common ratio is greater than 1, each term in the sequence will increase, leading to growth. Conversely, if the common ratio is between 0 and 1, the terms will decrease and approach zero. This relationship highlights how crucial understanding the common ratio is when analyzing sequences.
• Discuss how geometric sequences can be applied to model real-world phenomena.
  • Geometric sequences are useful in various real-world applications, such as modeling population growth, where each generation grows by a certain percentage, representing a fixed growth rate. They also apply to finance in calculating compound interest, where money grows exponentially based on an initial amount and a fixed interest rate over time. Additionally, they can represent processes like radioactive decay, showcasing how quantities diminish at consistent ratios.
• Evaluate the conditions under which an infinite geometric series converges and its implications for limits.
  • An infinite geometric series converges when the absolute value of its common ratio is less than 1. Under these conditions, the series approaches a finite limit, allowing us to sum it using the formula $$S = \frac{a_1}{1 - r}$$. This characteristic makes geometric sequences essential in calculus and analysis since understanding convergence helps determine the behavior of sequences at infinity, impacting various mathematical concepts.

Related terms

Common Ratio: The constant factor that each term in a geometric sequence is multiplied by to get the next term.

Convergence: The property of a sequence where the terms approach a specific value as they progress towards infinity.

Arithmetic Sequence: A sequence of numbers where each term after the first is obtained by adding a constant value to the previous term.