Geometric sequence

Definition

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. This concept is crucial as it connects to various mathematical principles, including how sequences can grow or shrink exponentially. Understanding geometric sequences is essential for solving problems that involve exponential growth or decay, and it forms a foundation for advanced mathematical concepts such as mathematical induction.

5 Must Know Facts For Your Next Test

1. In a geometric sequence, if the common ratio is between 0 and 1, the terms will decrease towards zero.
2. The sum of an infinite geometric series can converge to a finite value if the absolute value of the common ratio is less than 1.
3. Geometric sequences can be represented graphically as exponential functions, showing their rapid increase or decrease.
4. The n-th term of a geometric sequence can be calculated using the formula $$a_n = a_1 imes r^{(n-1)}$$.
5. Geometric sequences often model real-world phenomena such as population growth, financial investments, and radioactive decay.

Review Questions

• How does the concept of a common ratio affect the behavior of a geometric sequence?
  • The common ratio determines whether a geometric sequence increases or decreases. If the common ratio is greater than 1, the terms grow larger rapidly, demonstrating exponential growth. Conversely, if the common ratio is between 0 and 1, each subsequent term shrinks, approaching zero. This fundamental property influences many mathematical applications, including modeling real-world scenarios.
• Describe how you would use mathematical induction to prove a statement about geometric sequences.
  • To use mathematical induction on geometric sequences, start with a base case to show that the statement holds true for the first term. Then assume it holds for an arbitrary term $$n$$ and prove it for the next term $$n + 1$$ using the properties of geometric sequences. This approach shows that if it's true for one term, it must be true for all subsequent terms based on the defined common ratio and behavior of the sequence.
• Evaluate how geometric sequences can be applied in real-world scenarios such as finance or science.
  • Geometric sequences have significant applications in real-world scenarios like finance and science due to their ability to model exponential growth or decay. In finance, they are used to calculate compound interest over time; for instance, if you invest money with a certain interest rate, your investment grows in proportion to the previous amount. In science, they can represent processes like radioactive decay or population growth where quantities change at rates proportional to their current size. Understanding these applications helps bridge theoretical math with practical use.

Related terms

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

Exponential Growth: A rapid increase in quantity, represented by a geometric sequence where the common ratio is greater than 1.

Formula for the n-th Term: The expression used to find any term in a geometric sequence, typically written as $$a_n = a_1 imes r^{(n-1)}$$, where $$a_1$$ is the first term and $$r$$ is the common ratio.