We are tasked with expanding $\left(3 - 2x\right)\left(1 + \frac{x}{2}\right)^{n}$ and finding the value of $n$ such that the coefficient of $x$ is 7. Afterward, we will also determine the coefficient of $x^2$.

Step 1: Expand $\left(1 + \frac{x}{2}\right)^{n}$

We will use the binomial theorem to expand $\left(1 + \frac{x}{2}\right)^{n}$ up to the terms involving $x$ and $x^2$, since we're interested in the coefficients of $x$ and $x^2$. The binomial expansion is:

$\left(1 + \frac{x}{2}\right)^{n} = 1 + \binom{n}{1} \left(\frac{x}{2}\right) + \binom{n}{2} \left(\frac{x}{2}\right)^2 + \cdots$

This simplifies to:

$\left(1 + \frac{x}{2}\right)^{n} = 1 + n\frac{x}{2} + \frac{n(n-1)}{2} \left(\frac{x}{2}\right)^2 + \cdots$

$\left(1 + \frac{x}{2}\right)^{n} = 1 + \frac{n}{2}x + \frac{n(n-1)}{8}x^2 + \cdots$

Step 2: Multiply by $3 - 2x$

Now, we multiply the expansion of $\left(1 + \frac{x}{2}\right)^{n}$ by $3 - 2x$:

$(3 - 2x)\left(1 + \frac{n}{2}x + \frac{n(n-1)}{8}x^2 + \cdots\right)$

Distribute $3 - 2x$:

$= 3(1 + \frac{n}{2}x + \frac{n(n-1)}{8}x^2 + \cdots) - 2x(1 + \frac{n}{2}x + \frac{n(n-1)}{8}x^2 + \cdots)$

This simplifies to:

$= 3 + \frac{3n}{2}x + \frac{3n(n-1)}{8}x^2 - 2x - n x^2 + \cdots$

Step 3: Collect terms involving $x$

From the expansion, the terms involving $x$ are:

$\frac{3n}{2}x - 2x$

For the coefficient of $x$ to be 7, we set up the equation:

$\frac{3n}{2} - 2 = 7$

Solve for $n$:

$\frac{3n}{2} = 9$

$3n = 18$

$n = 6$

Step 4: Find the coefficient of $x^2$

Now that we know $n = 6$, substitute this into the terms involving $x^2$. The terms involving $x^2$ are:

$\frac{3n(n-1)}{8}x^2 - n x^2$

Substitute $n = 6$:

$\frac{3 \times 6 \times 5}{8}x^2 - 6x^2$

Simplify:

$\frac{90}{8}x^2 - 6x^2 = 11.25x^2 - 6x^2 = 5.25x^2$

Thus, the coefficient of $x^2$ is $5.25$.

Final Answer:

• The value of $n$ is $6$.
• The coefficient of $x^2$ is $5.25$.

Would you like further details on any step?

Here are five related questions for practice:

1. What is the binomial expansion of $\left(1 + \frac{x}{3}\right)^5$?
2. How would the expansion change if the expression was $(2 - x)\left(1 + \frac{x}{3}\right)^n$?
3. What is the general form of the binomial expansion for $(a + bx)^n$?
4. How do you find the coefficient of a specific power in a binomial expansion?
5. What are the properties of binomial coefficients?

Tip: In binomial expansions, identifying the terms you need early on can save a lot of time.