In a geometric sequence, the ratio of any two consecutive terms is constant. Let the given terms be $a = 2k + 7$, $b = 1 - k$, and $c = k - 7$. For these terms to be in a geometric sequence, the following condition must hold:

$\frac{b}{a} = \frac{c}{b}$

Step 1: Write the condition for the geometric sequence

$\frac{1 - k}{2k + 7} = \frac{k - 7}{1 - k}$

Cross-multiply: $(1 - k)^2 = (2k + 7)(k - 7)$

Step 2: Expand both sides

$(1 - k)^2 = 1 - 2k + k^2$ $(2k + 7)(k - 7) = 2k^2 - 14k + 7k - 49 = 2k^2 - 7k - 49$

Equating these: $1 - 2k + k^2 = 2k^2 - 7k - 49$

Step 3: Simplify the quadratic equation

Rearrange all terms to one side: $0 = 2k^2 - 7k - 49 - (1 - 2k + k^2)$ $0 = 2k^2 - 7k - 49 - 1 + 2k - k^2$ $0 = k^2 - 5k - 50$

Step 4: Solve the quadratic equation

The equation is: $k^2 - 5k - 50 = 0$

Using the quadratic formula: $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad a = 1, b = -5, c = -50$ $k = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-50)}}{2(1)}$ $k = \frac{5 \pm \sqrt{25 + 200}}{2}$ $k = \frac{5 \pm \sqrt{225}}{2}$ $k = \frac{5 \pm 15}{2}$

Thus: $k = \frac{5 + 15}{2} = 10 \quad \text{or} \quad k = \frac{5 - 15}{2} = -5$

Step 5: Find the common ratio $r$ for each case

1. Case 1: $k = 10$

   • Terms: $2k + 7 = 27$, $1 - k = -9$, $k - 7 = 3$
   • Common ratio: $r = \frac{b}{a} = \frac{-9}{27} = -\frac{1}{3}$

2. Case 2: $k = -5$

   • Terms: $2k + 7 = -3$, $1 - k = 6$, $k - 7 = -12$
   • Common ratio: $r = \frac{b}{a} = \frac{6}{-3} = -2$

Final Answer:

• For $k = 10$: $r = -\frac{1}{3}$
• For $k = -5$: $r = -2$

Would you like a more detailed explanation of any step? 😊

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Related Questions:

1. How can the quadratic formula be applied to other geometric sequences?
2. What are some real-world applications of geometric sequences?
3. How do the signs of terms in a geometric sequence affect the common ratio?
4. Can $k$ take complex values in this scenario? Why or why not?
5. What happens if the sequence is arithmetic instead of geometric?

Tip: Always double-check your solutions in sequences by recalculating the ratios to confirm the pattern!