Let's analyze the two questions presented in the image:

First Question:

Given the function:

$f(x) = \frac{x^3 + 3\sqrt{x}}{x^{3/2}}, \quad x > 0$

We need to simplify the expression to find $f(x)$ and match it with one of the given options.

First, let's simplify the expression:

$f(x) = \frac{x^3}{x^{3/2}} + \frac{3\sqrt{x}}{x^{3/2}}$

This can be broken down as follows:

1. Simplify the first term:

$\frac{x^3}{x^{3/2}} = x^{3 - 3/2} = x^{3/2}$

2. Simplify the second term:

$\frac{3\sqrt{x}}{x^{3/2}} = \frac{3x^{1/2}}{x^{3/2}} = \frac{3}{x^{2/2}} = \frac{3}{x} = 3x^{-1}$

So the function simplifies to:

$f(x) = x^{3/2} + 3x^{-1}$

Answer: The correct answer is $\boxed{B}$.

Second Question:

We are given that:

$a_0 = 1, \quad a_{n+1} = 2a_n + 1 \quad \text{for } n \geq 0$

We need to find the value of:

$\sum_{n=0}^{10} a_n$

To solve this, first observe the pattern in the sequence $a_n$.

The recursive relation is:

$a_{n+1} = 2a_n + 1$

Calculate the first few terms:

a_0 &= 1 \\ a_1 &= 2(1) + 1 = 3 \\ a_2 &= 2(3) + 1 = 7 \\ a_3 &= 2(7) + 1 = 15 \\ a_4 &= 2(15) + 1 = 31 \\ \end{aligned}$$ It appears that $$a_n = 2^{n+1} - 1$$. To find the sum $$\sum_{n=0}^{10} a_n$$: $$\sum_{n=0}^{10} a_n = \sum_{n=0}^{10} (2^{n+1} - 1)$$ $$= \sum_{n=0}^{10} 2^{n+1} - \sum_{n=0}^{10} 1$$ The first sum is a geometric series: $$\sum_{n=0}^{10} 2^{n+1} = 2\sum_{n=0}^{10} 2^n = 2 \times \frac{2^{11} - 1}{2 - 1} = 2(2048 - 1) = 4094$$ The second sum: $$\sum_{n=0}^{10} 1 = 11$$ Thus: $$\sum_{n=0}^{10} a_n = 4094 - 11 = 4083$$ **Answer**: The correct answer is $$\boxed{C}$$. --- Would you like a detailed breakdown or have any further questions? ### Relative Questions: 1. How does the concept of geometric series apply to recursive sequences? 2. Can you provide more examples of simplifying rational expressions? 3. How do recursive sequences relate to geometric progressions? 4. What are the general steps for simplifying complex fractions? 5. How do you identify the pattern in a recursive sequence? ### Tip: Always look for patterns or simplify expressions in recursive sequences to find closed-form solutions or sums.