Tribonacci Sequence Term 
A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its seeds For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $6$
  ($1+2+3$),then $11(2+3+6)$.

Find the smallest 5 digit term in a tribonacci sequence if the seeds are $6,19,22$
I'm having trouble with this. I don't know where to start. The formula for the tribonacci sequence in relation to its seeds is $$u_{n+3} = u_{n} + u_{n+1} + u_{n+2}$$
This tribonacci formula holds for all integer $n$. But that's all I know how to work out. And just if it helps, the next few numbers in the sequence mentioned in the question are $47,88,157,292$. Is there some shortcut to it, because I need to show some working out and having two pages full of addition doesn't sound very easy to mark, does it?

By the theory of linear recurrences, the sequence approximately follows a geometric progression
$$u_n=ar^n$$ where $r$ is the largest root of $r^3=r^2+r+1$, which is about $1.8392867552142$.
With $u_7=292$, we estimate $a=4.1005$.
Then we can expect $u_n\ge10000$ for 
$$n\ge\frac{\log 10000-\log a}{\log r}=12.798\cdots$$

Indeed,
$$1\to 6\\
2\to 19\\
3\to 22\\
4\to 47\\
5\to 88\\
6\to 157\\
7\to 292\\
8\to 537\\
9\to 986\\
10\to 1815\\
11\to 3338\\
12\to 6139\\
13\to 11292\\
$$
