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 describes a geometric sequence where the weekly allowance doubles every two weeks. The key components of the sequence are:

- **First term (a₁):** 1 cent (the allowance in the first week).
- **Doubling every two weeks:** This suggests the common ratio (r) is 2 for every two weeks.
- **Number of weeks (n):** 30 weeks.

However, the geometric sequence must account for the change in the allowance every two weeks. So we treat each "block" of two weeks as one term in the sequence.

### Step-by-Step Breakdown:
1. **Identify terms per two-week period:**
   Since the allowance doubles every two weeks, we divide the 30 weeks into **15 periods** (because 30 ÷ 2 = 15).

2. **Geometric sequence formula:**
   The formula for a geometric sequence is:
   \[
   a_n = a_1 \cdot r^{n-1}
   \]
   Where:
   - $a_n$ is the nth term,
   - $a_1$ is the first term (which is 1 cent),
   - $r$ is the common ratio (which is 2),
   - $n$ is the number of terms.

3. **Allowance after two-week periods:**
   For each two-week period, the allowance doubles. Therefore, the amount received every week in each period is the same within that period.

   - **First two weeks:** You get 1 cent per week.
   - **Next two weeks:** You get 2 cents per week.
   - **Next two weeks:** You get 4 cents per week.
   - **And so on...**

4. **Finding the total allowance:**
   The allowance sequence grows in powers of 2, starting from 1 cent. To calculate the total allowance over 30 weeks, we sum up the weekly allowance for each two-week period:

   \[
   S = 2(1 + 2 + 4 + 8 + \dots + 2^{14})
   \]
   This is a geometric series sum with:
   - First term $a = 1$,
   - Common ratio $r = 2$,
   - Number of terms $n = 15$.

5. **Geometric series sum formula:**
   The sum of the first $n$ terms of a geometric series is given by:
   \[
   S_n = a \cdot \frac{r^n - 1}{r - 1}
   \]
   Plugging in the values:
   \[
   S_{15} = 1 \cdot \frac{2^{15} - 1}{2 - 1} = 2^{15} - 1 = 32768 - 1 = 32767
   \]

6. **Multiply by 2 for each week:**
   Since the amount doubles every two weeks, the total allowance over the 30 weeks is:
   \[
   \text{Total allowance} = 2 \times 32767 = 65534 \text{ cents} = 655.34 \text{ dollars}.
   \]

### Final Answer:
The total allowance received over 30 weeks is **$655.34**.

Would you like more details or further clarifications?

Here are 5 follow-up questions to expand this understanding:
1. How would the result change if the allowance doubled every three weeks instead of two?
2. What is the total allowance if it triples every two weeks?
3. Can you find the total allowance for 40 weeks under the same conditions?
4. How would the sum change if the doubling started after the first week instead of the second?
5. How does the sum of a geometric series compare to an arithmetic one over the same period?

Tip: Understanding the common ratio and applying it to time intervals is key to solving geometric sequence problems!

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?

This problem explores how to calculate the total allowance over 30 weeks with a weekly allowance that doubles every two weeks, starting at 1 cent. Key concepts include geometric sequences and series summation, suitable for Grades 9-12.

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