We are given the expression:

$\left( 1 + \frac{2}{3x} \right)^n (3 + nx)^2 = 9 + 84x + \dots$

We need to find the value of $n$. Let's solve it step by step.

Step 1: Expand the terms

The expression involves two terms to expand:

1. $\left( 1 + \frac{2}{3x} \right)^n$
2. $(3 + nx)^2$

Step 2: Expand $\left( 1 + \frac{2}{3x} \right)^n$

We can use the binomial expansion to expand $\left( 1 + \frac{2}{3x} \right)^n$. The binomial expansion of $(1 + a)^n$ is given by:

$(1 + a)^n = 1 + n \cdot a + \frac{n(n-1)}{2!} \cdot a^2 + \cdots$

For $a = \frac{2}{3x}$, we get the following first few terms:

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

Step 3: Expand $(3 + nx)^2$

Now, expand $(3 + nx)^2$:

$(3 + nx)^2 = 9 + 6nx + n^2 x^2$

Step 4: Multiply the two expansions

Now, multiply the two expanded expressions:

$\left( 1 + n \cdot \frac{2}{3x} + \cdots \right) \cdot \left( 9 + 6nx + n^2 x^2 + \cdots \right)$

Let's focus on the terms up to $x$, since the right-hand side of the equation is given as $9 + 84x + \cdots$.

• The constant term: $1 \times 9 = 9$
• The $x$-term: $1 \times 6nx = 6nx$ and $n \cdot \frac{2}{3x} \times 9 = \frac{18n}{3x} = 6n$ (note that this term contributes to the coefficient of $x$).

Thus, the coefficient of $x$ from the product of the two terms is:

$6n + 6n = 12n$

We know that the coefficient of $x$ in the final expression is 84, so we set:

$12n = 84$

Step 5: Solve for $n$

Solve for $n$:

$n = \frac{84}{12} = 7$

Thus, the value of $n$ is $\boxed{7}$.

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Would you like further details on any of the steps, or do you have any questions about this solution?

Here are some related questions you can explore:

1. How do you perform a binomial expansion for higher-order terms?
2. What is the significance of each term in the binomial expansion?
3. How do you multiply two binomial expansions correctly?
4. How can we generalize the method for higher powers or different types of expressions?
5. Can you calculate the coefficient of $x^2$ in this problem?

Tip: When expanding binomials, keep track of each term and ensure you only combine like terms (e.g., powers of $x$).