b) $$ \sum_{i=1}^{n} 3 \cdot 2^{i-1}=12285 $$

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UpStudy AI · Accepted Answer

To solve the equation:

$$
\sum_{i=1}^{n} 3 \cdot 2^{i-1} = 12,\!285
$$

we can recognize that this is a geometric series. Here's how to approach it step-by-step:

1. **Identify the Components of the Geometric Series:**
   - **First term ($$a_1$$)**: When $$i = 1$$,
     $$
     a_1 = 3 \cdot 2^{1-1} = 3 \cdot 2^0 = 3 \cdot 1 = 3
     $$
   - **Common ratio ($$r$$)**: Each term is multiplied by 2 to get the next term, so $$r = 2$$.

2. **Write the Sum Formula for a Geometric Series:**
   The sum of the first $$n$$ terms of a geometric series is given by:
   $$
   S_n = a_1 \cdot \frac{r^n - 1}{r - 1}
   $$
   
   Plugging in the known values:
   $$
   12,\!285 = 3 \cdot \frac{2^n - 1}{2 - 1}
   $$
   Simplifies to:
   $$
   12,\!285 = 3 \cdot (2^n - 1)
   $$

3. **Solve for $$2^n$$:**
   - Divide both sides by 3:
     $$
     2^n - 1 = 4,\!095
     $$
   - Add 1 to both sides:
     $$
     2^n = 4,\!096
     $$

4. **Find $$n$$:**
   - Recognize that $$4,\!096$$ is a power of 2:
     $$
     2^{12} = 4,\!096
     $$
   - Therefore, $$n = 12$$.

**Final Answer:**  
The value of $$ n $$ is **12**.
$$ n = 12 $$

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