What is the geometric sequence used to calculate the allowance received during thirty weeks if you receive 1 cent a week and it doubles every two weeks

The problem involves a **geometric sequence** where the amount of allowance starts at 1 cent and doubles every two weeks over a period of 30 weeks.

### Steps to solve the problem:

1. **Identify the first term (a):**  
   The first term of the sequence is $ a = 0.01 $ (1 cent).

2. **Determine the common ratio (r):**  
   Since the allowance doubles every two weeks, the common ratio between every two terms is $ r = 2 $.

3. **Set the general formula for a geometric sequence:**
   The formula for the $n$-th term of a geometric sequence is:
   \[
   a_n = a \cdot r^{\frac{n}{k}}
   \]
   Where:
   - $ a $ is the first term,
   - $ r $ is the common ratio,
   - $ n $ is the number of weeks,
   - $ k $ is the period in which the doubling occurs (in this case, 2 weeks).

   In this situation, the formula becomes:
   \[
   a_n = 0.01 \cdot 2^{\frac{n}{2}}
   \]
   This formula gives the allowance received on week $n$, where the allowance doubles every 2 weeks.

### Example: Calculate allowance in week 30

To calculate the allowance received in week 30:
\[
a_{30} = 0.01 \cdot 2^{\frac{30}{2}} = 0.01 \cdot 2^{15} = 0.01 \cdot 32768 = 327.68 \, \text{cents} = 3.2768 \, \text{dollars}
\]

So, the allowance in week 30 is approximately $3.28.

### Total Allowance over 30 weeks:

To calculate the total allowance over 30 weeks, you sum all the terms of the geometric sequence from week 1 to week 30. This is a finite geometric series, and the sum formula for a geometric series is:
\[
S_n = a \cdot \frac{1 - r^n}{1 - r}
\]
Where:
- $ S_n $ is the sum of the first $ n $ terms,
- $ a = 0.01 $ is the first term,
- $ r = 2^{\frac{1}{2}} $ is the common ratio every week,
- $ n = 30 $ is the number of weeks.

Would you like to proceed with the total sum calculation or any other details?

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Here are some related questions to explore:

1. What happens to the allowance if it triples every two weeks?
2. How would the sequence change if the allowance doubles every week instead of every two weeks?
3. Can you derive the total allowance if the starting allowance is 2 cents instead of 1 cent?
4. How can this geometric sequence be modeled using logarithms?
5. What would be the allowance at week 60 under the same doubling pattern?

**Tip:** For problems involving exponential growth, like this one, geometric sequences are a key tool to model the process!

What is the geometric sequence used to calculate the allowance received during thirty weeks if you receive 1 cent a week and it doubles every two weeks?

Learn how to calculate the allowance received over 30 weeks using a geometric sequence where the amount doubles every two weeks. Explore key concepts like geometric sequences and exponential growth, with detailed steps to calculate the total allowance.

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