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.

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.

UpStudy AI · Accepted Answer

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
The positive constant ratio is 2.

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.

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