Given that the first three terms of a geometric sequence are $(k + 4)$, $k$, and $(2k - 15)$, where $k$ is a positive constant:

Part (i) Show that $k = 12$

In a geometric sequence, the ratio between successive terms is constant. Therefore, the ratio between the first and second terms must be equal to the ratio between the second and third terms.

$\frac{k}{k + 4} = \frac{2k - 15}{k}$

Cross-multiply to solve for $k$:

$k \cdot k = (k + 4)(2k - 15)$

Expanding both sides:

$k^2 = 2k^2 - 15k + 8k - 60$

Simplify:

$k^2 = 2k^2 - 7k - 60$

Move all terms to one side:

$0 = k^2 - 7k - 60$

This is a quadratic equation. We can solve for $k$ using the quadratic formula:

$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 1$, $b = -7$, and $c = -60$:

$k = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(-60)}}{2(1)}$

$k = \frac{7 \pm \sqrt{49 + 240}}{2}$

$k = \frac{7 \pm \sqrt{289}}{2}$

$k = \frac{7 \pm 17}{2}$

This gives two possible values for $k$:

$k = \frac{24}{2} = 12 \quad \text{or} \quad k = \frac{-10}{2} = -5$

Since $k$ is a positive constant, $k = 12$.

Part (ii) Find the Common Ratio

Substitute $k = 12$ back into the expressions for the first three terms:

• First term: $k + 4 = 12 + 4 = 16$
• Second term: $k = 12$
• Third term: $2k - 15 = 2(12) - 15 = 24 - 15 = 9$

The common ratio $r$ is:

$r = \frac{\text{Second term}}{\text{First term}} = \frac{12}{16} = \frac{3}{4}$

You can also confirm this with the ratio between the third and second terms:

$r = \frac{\text{Third term}}{\text{Second term}} = \frac{9}{12} = \frac{3}{4}$

So, the common ratio $r = \frac{3}{4}$.

Part (iii) Find the 10th Term

The formula for the $n$th term of a geometric sequence is:

$a_n = a_1 \cdot r^{n-1}$

Where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. For the 10th term:

$a_{10} = 16 \cdot \left(\frac{3}{4}\right)^{9}$

Let's calculate this:

$a_{10} = 16 \cdot \left(\frac{3^9}{4^9}\right)$

$a_{10} = 16 \cdot \frac{19683}{262144}$

$a_{10} \approx 16 \cdot 0.0751 = 1.2016$

So, the 10th term is approximately $1.2016$.

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Would you like more details or have any other questions?

Here are 5 related questions for further practice:

1. What is the sum of the first 10 terms of this geometric sequence?
2. How would the common ratio change if the first term were doubled?
3. What would the sequence look like if the common ratio were negative?
4. How can you derive the sum to infinity for this sequence?
5. What is the relationship between the common ratio and the rate of convergence of the sequence?

Tip: For geometric sequences, the common ratio can significantly influence the behavior of the sequence, especially in determining whether the terms converge or diverge.