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?

In this paper, the authors show that, if $S_1,S_2,S_3$ are the seeds, then 
$$S_n=T_{n-2}\,S_1+(T_{n-2}+T_{n-3})\,S_2+T_{n-1}\,S_3$$ where $T_k$ is the "usual" Tribonacci number.
Applied to the seeds you give, this generates the following values
$$\left(
\begin{array}{ccc}
n & T_n & S_n \\
 1 & 0 & 6 \\
 2 & 1 & 19 \\
 3 & 1 & 22 \\
 4 & 2 & 47 \\
 5 & 4 & 88 \\
 6 & 7 & 157 \\
 7 & 13 & 292 \\
 8 & 24 & 537 \\
 9 & 44 & 986 \\
 10 & 81 & 1815
\end{array}
\right)$$
Hoping that this could help. Just continue for a few terms to get the answer.
Edit
In this paper, the author shows that $$\lim_{n\to \infty } \, \frac{S_{n+1}}{S_{n}}=\lim_{n\to \infty } \, \frac{T_{n+1}}{T_{n}}=\tau=\frac{1}{3} \left(1+\sqrt[3]{19-3 \sqrt{33}}+\sqrt[3]{19+3 \sqrt{33}}\right)$$ which is $\approx 1.83929$. This could also help you to find your result.
Using the last term you provided, making the approximation $S_n=\text{Round}\left[292 \tau ^{n-7}\right]$, the next terms would be $537, 988, 1817$.
