Learning Task 3. Tell something about what you have learned by answeringthe following questions.1. Explain how to determine the next term of the geometric sequence 3, 6, 12,24,2. In your opinion, where can you apply the concepts and skills that youlearned about geometric sequences? Explain.3. What are the differences between an arithmetic sequence and a geometricsequence?
Answer (1)
Answer:Learning Task 3: Geometric Sequences1. Determining the Next TermA geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value called the common ratio. To find the next term in the sequence 3, 6, 12, 24:Identify the common ratio: Divide any term by its preceding term. For example, 6 / 3 = 2, 12 / 6 = 2, and 24 / 12 = 2. The common ratio is 2.Multiply the last term by the common ratio: 24 * 2 = 48.Therefore, the next term in the sequence is 48.2. Applications of Geometric SequencesGeometric sequences have applications in various fields:Compound Interest: The growth of money invested with compound interest follows a geometric sequence. Each period, the interest earned is added to the principal, and the next period's interest is calculated on the new, larger principal.Population Growth: In some cases, population growth can be modeled using a geometric sequence, especially when resources are abundant, and the population grows at a constant rate.Radioactive Decay: The decay of radioactive substances follows a geometric sequence. The amount of radioactive material decreases by a fixed proportion at regular intervals.Computer Science: Geometric sequences can be used to analyze algorithms and data structures, such as binary search, where the search space is halved with each step.3. Differences between Arithmetic and Geometric SequencesFeature Arithmetic Sequence Geometric SequencePattern Each term is found by adding a constant value (common difference) to the previous term. Each term is found by multiplying the previous term by a constant value (common ratio).Formula a<sub>n</sub> = a<sub>1</sub> + (n - 1)d a<sub>n</sub> = a<sub>1</sub> * r<sup>(n-1)</sup>Example 2, 5, 8, 11... (common difference = 3) 3, 9, 27, 81... (common ratio = 3)In summary:Geometric sequences are characterized by a constant multiplier (common ratio) between consecutive terms.They have applications in various fields, including finance, biology, and computer science.The key difference between arithmetic and geometric sequences lies in the operation used to generate the next term: addition for arithmetic and multiplication for geometric.Step-by-step explanation:
