1. Problem 6: Given that $P=3$ when $n=1$, determine which equation might represent the pattern. Check each option by substituting $n=1$: - a) $P=3n \Rightarrow P=3\times1=3$ (matches) - b) $P=n+3 \Rightarrow P=1+3=4$ (no) - c) $P=2n+1 \Rightarrow P=2\times1+1=3$ (matches) - d) $P=3-n \Rightarrow P=3-1=2$ (no) So options a) and c) fit the initial condition. 2. Problem 9: Find the equation relating number of squares $s$ to figure number $f$ given the table: $f$: 1, 2, 3, 4, 5 $s$: 5, 7, 9, 11, 13 Observe the pattern: $s$ increases by 2 each time. Try each option: - a) $s=4f+1$; for $f=1$, $s=5$ correct, but for $f=2$, $s=9$ (not 7), no. - b) $s=2f+3$; for $f=1$, $s=5$ correct, $f=2$, $s=7$ correct, matches all. - c) $s=f+2$; for $f=1$, $s=3$ no. - d) $f=2s+3$; rearranged, not matching pattern. Answer: b) $s=2f+3$. 3. Problem 12: (a) Table of values for figure number $n$ and toothpicks $t$: - Figure 1: 3 toothpicks (one triangle) - Figure 2: 5 toothpicks (two triangles sharing one side) - Figure 3: 7 toothpicks - Figure 4: 9 toothpicks Pattern: each new figure adds 2 toothpicks. (b) Expression for toothpicks: $$t=2n+1$$ (c) Number of toothpicks in figure 45: $$t=2\times45+1=90+1=91$$ (d) Equation relating $t$ to $n$ is $t=2n+1$. (e) Which figure has 17 toothpicks? Solve $17=2n+1$: $$2n=16 \Rightarrow n=8$$ Figure 8 has 17 toothpicks. To check, substitute $n=8$ back into the equation. Final answers: - Problem 6: Equations a) and c) fit. - Problem 9: Equation b) fits. - Problem 12: $t=2n+1$, figure 45 has 91 toothpicks, figure 8 has 17 toothpicks.