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The problem you provided asks to find the sum to infinity of a geometric progression (GP) where the fifth, sixth, and seventh terms are given as \(8k\), \(-12\), and \(2k\), respectively. We are also told that \(k\) is negative.

### Step 1: Understanding the properties of a geometric progression
In a geometric progression, consecutive terms are related by a common ratio \(r\). Therefore, for any three consecutive terms, we have the following relationship:
\[
T_6 = r \cdot T_5 \quad \text{and} \quad T_7 = r \cdot T_6
\]
Given:
- \(T_5 = 8k\)
- \(T_6 = -12\)
- \(T_7 = 2k\)

### Step 2: Establishing the ratio
We can use the relation between consecutive terms to establish the common ratio \(r\).
\[
r = \frac{T_6}{T_5} = \frac{-12}{8k} = \frac{-3}{2k}
\]
Also, from \(T_7\) and \(T_6\):
\[
r = \frac{T_7}{T_6} = \frac{2k}{-12} = \frac{-k}{6}
\]

### Step 3: Solving for \(k\)
We now equate the two expressions for \(r\):
\[
\frac{-3}{2k} = \frac{-k}{6}
\]
Multiplying both sides by \(6k\) to eliminate the denominators:
\[
-18 = -k^2
\]
\[
k^2 = 18
\]
\[
k = -\sqrt{18} = -3\sqrt{2}
\]
(Note that \(k\) is negative, so we take the negative root.)

### Step 4: Finding the common ratio \(r\)
Substitute \(k = -3\sqrt{2}\) into one of the expressions for \(r\):
\[
r = \frac{-3}{2k} = \frac{-3}{2(-3\sqrt{2})} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}
\]

### Step 5: Sum to infinity of the geometric progression
The sum to infinity of a geometric progression is given by the formula:
\[
S_{\infty} = \frac{a}{1 - r}
\]
where \(a\) is the first term and \(r\) is the common ratio. We don't know \(a\) explicitly, but since the ratio \(r = \frac{\sqrt{2}}{4}\) is less than 1, the sum converges. To find the exact value, we would need the first term, which isn't directly given.

Would you like more details or help with further steps?

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Here are 5 related questions:
1. How do you derive the formula for the sum to infinity of a geometric progression?
2. Can the common ratio of a geometric progression be greater than 1 for the series to converge?
3. What are the conditions for a geometric progression to have a finite sum to infinity?
4. How does the value of \(k\) affect the progression terms and their properties?
5. What is the significance of knowing that \(k\) is negative in this problem?

**Tip:** Always check if the common ratio \(r\) is less than 1 for the sum to infinity to converge in a geometric progression.

The fifth, sixth and seventh terms of a geometric progression are 8k, -12 and 2k respectively. Given that k is negative, find the sum to infinity of the progression.

Learn how to find the sum to infinity of a geometric progression where the fifth, sixth, and seventh terms are given as 8k, -12, and 2k, respectively. The solution involves key concepts in algebra and geometric progressions, including the calculation of the common ratio and solving for k using a quadratic equation.

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