Question
The sum of the first 4 terms of a geometric series is 15 and the sum of the next 4 terms is 240 . Determine the positive constant ratio.
Ask by Daniel Cole. in South Africa
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The positive constant ratio is 2.
Solution
Let’s solve the problem step-by-step.
**1. Define the Variables:**
- Let \( a \) be the first term of the geometric series.
- Let \( r \) be the common ratio (which is positive).
**2. Express the Sums:**
- The sum of the first 4 terms is given by:
\[
S_4 = a + ar + ar^2 + ar^3 = a(1 + r + r^2 + r^3) = 15
\]
- The sum of the next 4 terms (terms 5 to 8) is:
\[
S_{next\ 4} = ar^4 + ar^5 + ar^6 + ar^7 = ar^4(1 + r + r^2 + r^3) = 240
\]
**3. Relate the Two Sums:**
Notice that:
\[
S_{next\ 4} = ar^4(1 + r + r^2 + r^3) = r^4 \cdot S_4
\]
Given that \( S_4 = 15 \):
\[
r^4 \cdot 15 = 240
\]
**4. Solve for \( r \):**
\[
r^4 = \frac{240}{15} = 16
\]
\[
r = \sqrt[4]{16} = 2
\]
**Conclusion:**
The positive constant ratio is **2**.
Answer: 2
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
In a geometric series, the sum of the first \( n \) terms can be calculated using the formula: \[ S_n = a \frac{1 - r^n}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. For the first 4 terms, we have: \[ S_4 = a \frac{1 - r^4}{1 - r} = 15 \] For the next 4 terms (the 5th to the 8th terms), the sum can be represented as: \[ S_{5-8} = ar^4 \frac{1 - r^4}{1 - r} = 240 \] Substituting \( S_4 \) into the equation for \( S_{5-8} \), we find: \[ S_{5-8} = r^4 \cdot 15 = 240 \] Solving for \( r^4 \): \[ r^4 = \frac{240}{15} = 16 \] Now, taking the fourth root of both sides gives us: \[ r = \sqrt[4]{16} = 2 \] So, the positive constant ratio is \( r = 2 \).
