The third term in the expansion of $(x + a)^n$ is $\binom{n}{2}x^2a^{n-2}$.
The fourth term is $\binom{n}{3}x^3a^{n-3}$.
The fifth term is $\binom{n}{4}x^4a^{n-4}$.
We are given that the third, fourth, and fifth terms are 84, 280, and 560 respectively.
So we have the following equations:
$\binom{n}{2}x^2a^{n-2} = 84$,
$\binom{n}{3}x^3a^{n-3} = 280$, and
$\binom{n}{4}x^4a^{n-4} = 560$.
Dividing the second equation by the first equation, we get $\frac{\binom{n}{3}x}{\binom{n}{2}} = \frac{280}{84} = \frac{10}{3}$.
Simplifying, we have $\frac{(n-2)x}{3} = \frac{10}{3}$.
Dividing the third equation by the second equation, we get $\frac{\binom{n}{4}x^4}{\binom{n}{3}x^3} = \frac{560}{280} = 2$.
Simplifying, we have $\frac{(n-3)x}{4} = 2$.
Now we can solve these two equations to find the value of $n$.
From the first equation, we have $(n-2)x = 10$.
From the second equation, we have $(n-3)x = 8$.
Dividing these two equations, we get $\frac{n-2}{n-3} = \frac{10}{8} = \frac{5}{4}$.
Cross-multiplying, we have $4(n-2) = 5(n-3)$.
Expanding, we get $4n - 8 = 5n - 15$.
Solving for $n$, we get $n = \boxed{7}$.The answer is: 7
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
If the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560 respectively, find the value of $n$.

### Answer:
