Let's work through the problems step by step:

Question 9:

The general formula for the sequence is $U_n = n^2 + 1$.

To find the first five terms, we plug values for $n = 1, 2, 3, 4, 5$ into the formula:

1. $U_1 = 1^2 + 1 = 2$
2. $U_2 = 2^2 + 1 = 5$
3. $U_3 = 3^2 + 1 = 10$
4. $U_4 = 4^2 + 1 = 17$
5. $U_5 = 5^2 + 1 = 26$

Thus, the first five terms are: 2, 5, 10, 17, 26.

So, the correct answer is C. 2, 5, 10, 17, 26.

Question 10:

The given sequence is 5, 9, 13, 17, ...

This is an arithmetic sequence with a common difference of 4. The general term for an arithmetic sequence is given by: $U_n = U_1 + (n-1) \cdot d$ where $U_1$ is the first term and $d$ is the common difference.

For this sequence:

• $U_1 = 5$
• $d = 4$

Thus, the nth term will be: $U_n = 5 + (n-1) \cdot 4 = 5 + 4n - 4 = 4n + 1$

So, the correct answer is B. $U_n = 1 + 4n$.

Question 11:

For a geometric sequence, the formula for the nth term is given by: $U_n = U_1 \cdot r^{(n-1)}$ Where:

• $U_1 = 8$
• $U_6 = \frac{1}{4}$

We need to find the common ratio $r$. The relationship between $U_1$ and $U_6$ is: $U_6 = 8 \cdot r^{(6-1)} = 8 \cdot r^5 = \frac{1}{4}$ Solving for $r$: $8 \cdot r^5 = \frac{1}{4}$ $r^5 = \frac{1}{32}$ Taking the 5th root of both sides: $r = \frac{1}{2}$

So, the correct answer is E. $r = \frac{1}{2}$.

Question 12:

The sequence is 32, 16, 8, 4, ...

This is a geometric sequence, and we can find the ratio by dividing the second term by the first term: $\frac{16}{32} = \frac{1}{2}$

So, the ratio is $\frac{1}{2}$.

The correct answer is E. $\frac{1}{2}$.

Question 13:

We need to solve for $x$ in the equation: $\frac{4x - 1}{6} + x = \frac{2}{3} \cdot \frac{1 - 3x}{2}$ Start by simplifying both sides:

• Left-hand side: $\frac{4x - 1}{6} + x = \frac{4x - 1}{6} + \frac{6x}{6} = \frac{4x - 1 + 6x}{6} = \frac{10x - 1}{6}$
• Right-hand side: $\frac{2}{3} \cdot \frac{1 - 3x}{2} = \frac{1 - 3x}{3}$

Now, set the two expressions equal: $\frac{10x - 1}{6} = \frac{1 - 3x}{3}$ Cross-multiply to eliminate the denominators: $3(10x - 1) = 6(1 - 3x)$ Simplify both sides: $30x - 3 = 6 - 18x$ Move all terms involving $x$ to one side: $30x + 18x = 6 + 3$ $48x = 9$ Solve for $x$: $x = \frac{9}{48} = \frac{3}{16}$

So, the correct answer is None of the provided options.

Would you like more details or clarifications on any of these steps? Let me know if you'd like me to explain further!

Related questions:

1. How do you identify whether a sequence is arithmetic or geometric?
2. How do you calculate the common ratio of a geometric sequence?
3. How do you solve for $x$ in a linear equation with fractions?
4. Can you explain the 5th root in the geometric sequence question?
5. What’s the general formula for the nth term in an arithmetic sequence?

Tip: When working with sequences, always first identify whether the sequence is arithmetic (constant difference) or geometric (constant ratio). This can help you apply the right formulas.