Geometric Sequence

Definition

A geometric sequence is a list 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 can be represented in a variety of mathematical contexts, including exponential growth or decay, which makes it important in fields like finance and science. The structure of a geometric sequence allows for the easy calculation of any term, as well as the sum of the terms in a series.

5 Must Know Facts For Your Next Test

1. In a geometric sequence, if the first term is 'a' and the common ratio is 'r', the nth term can be expressed as $$a \cdot r^{(n-1)}$$.
2. The sum of the first n terms of a geometric sequence can be calculated using the formula $$S_n = a \frac{1 - r^n}{1 - r}$$ if $$r \neq 1$$.
3. Geometric sequences can have positive or negative common ratios, leading to different behaviors in their terms (e.g., alternating signs if the common ratio is negative).
4. If the absolute value of the common ratio is less than 1, the terms of the geometric sequence will approach zero as n increases.
5. In finance, geometric sequences are often used to calculate compound interest, where interest accumulates on both the principal and previously accrued interest.

Review Questions

• How can you determine whether a given sequence is geometric, and what role does the common ratio play in this determination?
  • To determine if a sequence is geometric, check if there is a consistent common ratio between consecutive terms. This means dividing each term by the preceding term should yield the same result across all pairs. If this holds true, then it's geometric. The common ratio not only defines how each term relates to its predecessor but also affects how quickly or slowly the terms grow or shrink in value.
• Explain how the formula for the sum of a finite geometric series can be derived from its definition and how it applies in practical scenarios.
  • The formula for the sum of a finite geometric series is derived by multiplying the sum by the common ratio and then subtracting this from the original sum. This leads to a simplified expression that reveals that you can capture all terms effectively with just one equation. This formula is practical in finance for calculating total returns on investments over time when compounded geometrically.
• Evaluate how understanding geometric sequences can enhance your ability to model real-world situations involving growth or decay.
  • Understanding geometric sequences allows you to model various real-world phenomena such as population growth, radioactive decay, and compound interest. By applying this knowledge, you can predict future values based on current data and manipulate variables like initial amounts and growth rates. This analytical skill not only enhances problem-solving capabilities but also provides insights into exponential changes that occur in many fields, leading to informed decision-making.

Related terms

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

Exponential Growth: A situation where the quantity increases at a rate proportional to its current value, often represented by geometric sequences.

Sum of a Geometric Series: The total value obtained by adding all terms of a geometric sequence, which can be calculated using specific formulas depending on whether the series is finite or infinite.