https://file.aiquickdraw.com/mathbot/file/172825703278618w3zg63.jpg

Let's break down the math problems visible in the image:

### Problem 14:
The sequence provided is $1, 3, 9, \dots$. It appears to be a geometric sequence where each term is multiplied by 3.

- **Recursive Formula**:  
  $ f(n) = f(n-1) \times 3 $, with $ f(1) = 1 $.
  
- **Explicit Formula**:  
  The explicit formula for a geometric sequence is $ f(n) = a \times r^{n-1} $, where $a$ is the first term and $r$ is the common ratio.  
  In this case, $ a = 1 $ and $ r = 3 $, so:  
  $ f(n) = 1 \times 3^{n-1} $, or simply,  
  $ f(n) = 3^{n-1} $.

### Problem 15:
The given terms are $4, 8, 16, \dots$, which also follow a geometric pattern, where each term is multiplied by 2.

- **Recursive Formula**:  
  $ f(n) = f(n-1) \times 2 $, with $ f(1) = 4 $.

- **Explicit Formula**:  
  Using the same formula for geometric sequences $ f(n) = a \times r^{n-1} $, where $ a = 4 $ and $ r = 2 $:  
  $ f(n) = 4 \times 2^{n-1} $.

### Problem 16:
This problem describes Shannon's savings, where she starts with $25 and deposits $15 each week.

- **Recursive Formula**:  
  $ f(n) = f(n-1) + 15 $, with $ f(0) = 25 $ (since at week 0, she had deposited $25).

- **Explicit Formula**:  
  The explicit formula for this arithmetic sequence can be written as:  
  $ f(n) = 15n + 25 $, where $ n $ is the number of weeks, and she starts with $25.

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Would you like further details or have any questions? Here are some related questions to consider:

1. How do we determine if a sequence is arithmetic or geometric?
2. How does the recursive formula relate to the explicit formula?
3. Can geometric sequences have ratios less than 1?
4. How would you graph the explicit formulas for these sequences?
5. What is the meaning of the initial condition in recursive formulas?

**Tip:** Explicit formulas allow you to find any term in the sequence directly, while recursive formulas depend on previous terms.

What are the recursive and explicit formulas for the sequences 1, 3, 9,... ; 4, 8, 16,... ; and Shannon's savings account scenario?

Explore the recursive and explicit formulas for geometric sequences such as 1, 3, 9,... and 4, 8, 16,... as well as arithmetic sequences like Shannon's savings account scenario. This includes key concepts such as geometric ratio and arithmetic difference, suitable for students in Grades 8-10.

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