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.

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