date: total toothpicks
1 toothpicks:3
2 toothpicks:9
3 toothpicks:18
4 toothpicks:30
5 toothpicks:45
6 toothpicks:63
I) model the toothpick column with a recursive relationship with $\displaystyle t_1=3$
Notice the pattern
$\displaystyle \begin{array}{r}
3 + 6 = 9\;\,\\
9 + 9 = 18\\
18 + 12 = 30 \\
30 + 15 = 45\\
45 + 18= 63\\
\end{array}$
or
$\displaystyle \begin{array}{r}
3 + 2 \cdot3 = 9\;\,\\
9 + 3 \cdot3 = 18\\
18 + 4 \cdot3 = 30 \\
30 + 5 \cdot3 = 45\\
45 + 6 \cdot3 = 63\\
\end{array}
$
This help?