Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( n = 12 \)
Solution
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**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve the equation \( \sum_{i=1}^{n} 3 \cdot 2^{i-1}=12285 \), we can first simplify the summation. It's a geometric series, where the first term \( a = 3 \) and the common ratio \( r = 2 \). The sum of the first \( n \) terms of a geometric series can be expressed as: \[ S_n = a \frac{r^n - 1}{r - 1} \] Plugging in our values: \[ S_n = 3 \frac{2^n - 1}{2 - 1} = 3(2^n - 1) \] Setting it equal to 12285: \[ 3(2^n - 1) = 12285 \] Dividing both sides by 3 gives: \[ 2^n - 1 = 4095 \] Adding 1 to both sides: \[ 2^n = 4096 \] Now, recognizing that \( 4096 = 2^{12} \), we find: \[ n = 12 \] Thus, the value of \( n \) is \( \boxed{12} \).
