1. **Problem Statement:** Keamogetse makes shapes using matches as shown in Figures 1 to 4. We need to analyze the number of matches used in each figure and find patterns and formulas. 2. **Matches in each figure (1.1):** - Figure 1: 4 matches - Figure 2: 7 matches - Figure 3: 10 matches - Figure 4: 13 matches 3. **Matches added each time (1.2):** From one figure to the next, the number of matches increases by $7 - 4 = 3$. 4. **Constant difference (1.3):** The constant difference in the number of matches is $3$. 5. **Multiply figure number by constant difference (1.4):** - $T_1 = 1 \times 3 = 3$ - $T_2 = 2 \times 3 = 6$ - $T_3 = 3 \times 3 = 9$ 6. **Relationship in words (1.5):** The number of matches increases by 3 for each additional figure number, starting from 4 matches at figure 1. 7. **Algebraic rule (general term) (1.6):** Since the first figure has 4 matches and each figure adds 3 matches, the formula is: $$T_n = 3n + 1$$ 8. **Matches in 32nd figure (1.7):** $$T_{32} = 3 \times 32 + 1 = 96 + 1 = 97$$ 9. **Figure number for 40 matches (1.8):** Solve for $n$: $$3n + 1 = 40$$ $$3n = 39$$ $$n = \frac{39}{3} = 13$$ 10. **Does any figure use exactly 59 matches? (1.9):** Solve for $n$: $$3n + 1 = 59$$ $$3n = 58$$ $$n = \frac{58}{3} = 19.33...$$ Since $n$ must be a whole number, no figure uses exactly 59 matches. **Final answers:** - 1.1: 4, 7, 10, 13 - 1.2: 3 - 1.3: 3 - 1.4: $T_1=3$, $T_2=6$, $T_3=9$ - 1.5: Number of matches increases by 3 per figure starting at 4 - 1.6: $T_n=3n+1$ - 1.7: 97 matches - 1.8: Figure number 13 - 1.9: No figure has exactly 59 matches