https://file.aiquickdraw.com/mathbot/file/17282574974058n914sdc.jpg

Let's analyze and solve the provided problems from the image:

### Problem 18: Recursive Sequence Analysis
Given:
- Recursive rule: \( f(n) = f(n-1) + 8 \)
- Initial value: \( f(1) = -10 \)

We need to evaluate which of the provided statements are valid based on the given sequence rules.

1. **Calculate a few terms of the sequence:**
   - \( f(1) = -10 \)
   - \( f(2) = f(1) + 8 = -10 + 8 = -2 \)
   - \( f(3) = f(2) + 8 = -2 + 8 = 6 \)
   - \( f(4) = f(3) + 8 = 6 + 8 = 14 \)
   - \( f(5) = f(4) + 8 = 14 + 8 = 22 \), and so on.

2. **Check the given options:**
   - **A. \( f(n) = 8n - 2 \)**: This is not correct because the initial value and the recursive rule do not match this expression.
   - **B. \( f(2) = -2 \)**: This is correct, as calculated above.
   - **C. (2,4)**: This suggests that \( f(2) = 4 \), which is incorrect. \( f(2) = -2 \).
   - **D. \( f(7) = -38 \)**: Need to calculate \( f(7) \). Using \( f(n) = f(1) + 8(n-1) \), \( f(7) = -10 + 8(6) = -10 + 48 = 38 \), not \(-38\). So, this is incorrect.
   - **E. \( f(n) = f(n-1) + 8, f(0) = -18 \)**: This is not mentioned as part of the sequence definition.
   - **F. \( f(n) = 8n - 18 \)**: The formula for an arithmetic sequence \( f(n) \) should be \( f(n) = f(1) + (n - 1)d \), where \( d = 8 \). Calculating for \( f(n) = -10 + 8(n - 1) \) gives a formula closer to \( 8n - 18 \). So, this might match.
   - **G. (4, 14)**: This is correct as \( f(4) = 14 \).
   - **H. \( f(10) = 62 \)**: Calculate \( f(10) \): \( f(10) = -10 + 8(9) = -10 + 72 = 62 \). This is correct.
   - **I. Sequence table calculation**: Looks to be consistent with the pattern.

### Problem 19: Arithmetic Sequences
1. **Find missing terms and define equations:**

   - Sequence 1: \( 4, 12, \_, 108, 324 \)
     - This is a geometric sequence with a ratio of 3. \( 12 \times 3 = 36 \), so the missing term is \( 36 \).

   - Sequence 2: \( 50, 35, 20, 5, \_, -25, -40 \)
     - This is an arithmetic sequence with a common difference of \(-15\). Thus, the missing term is \( -10 \).

2. **Recursive and Explicit Equations:**
   - **Sequence 1 (Geometric)**
     - Recursive: \( f(n) = f(n-1) \times 3, f(1) = 4 \)
     - Explicit: \( f(n) = 4 \times 3^{n-1} \)
     - Graph shape: Exponential growth (non-linear)

   - **Sequence 2 (Arithmetic)**
     - Recursive: \( f(n) = f(n-1) - 15, f(1) = 50 \)
     - Explicit: \( f(n) = 50 - 15(n-1) \)
     - Graph shape: Linear

3. **Calculate the 10th term:**
   - **Geometric sequence:** \( f(10) = 4 \times 3^9 = 4 \times 19683 = 78732 \).
   - **Arithmetic sequence:** \( f(10) = 50 - 15 \times 9 = 50 - 135 = -85 \).

Would you like further details on any part of this solution or explanation? 

### Follow-up Questions
1. How do you find the general formula for a recursive sequence?
2. What is the difference between an arithmetic and a geometric sequence?
3. How can you determine the common difference or ratio in a sequence?
4. Why does the geometric sequence graph appear as exponential growth?
5. What would the graph of a recursively defined sequence look like compared to its explicit formula?

### Tip
In an arithmetic sequence, the difference between consecutive terms remains constant, while in a geometric sequence, the ratio between consecutive terms stays constant.

Given the recursive rule f(n) = f(n-1) + 8, f(1) = -10, and the arithmetic sequences, solve for the missing terms and analyze the recursive sequences.

Analyze recursive and arithmetic sequences with a step-by-step solution to finding missing terms and analyzing recursive rules. This math problem involves sequences, recursive formulas, and explicit equations for both arithmetic and geometric sequences.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation