1. The problem asks for a rule for the number of sticks in the nth pattern, given: 1st pattern has 5 sticks, 2nd pattern has 9 sticks, 3rd pattern has 13 sticks. 2. Observe the pattern: from 5 to 9, increase by 4; from 9 to 13, increase by 4 again. 3. This suggests a linear pattern with a common difference of 4. 4. The general form for a linear pattern is $$a_n = a_1 + (n-1)d$$ where $a_n$ is the nth term, $a_1$ is the first term, $d$ is the difference. 5. Substitute $a_1=5$ and $d=4$: $$a_n = 5 + (n-1)4$$ 6. Simplify: $$a_n = 5 + 4n - 4 = 4n + 1$$ 7. So, the rule for the number of sticks in the nth pattern is $$a_n = 4n + 1$$.