1. **Problem Statement:** We have a sequence of patterns made of matchsticks forming triangles. The first pattern has 3 matchsticks, the second 5, the third 7, and so on. We need to find: a) The number to fill in the box for the term-to-term rule. b) The number of matchsticks in the 4th pattern. 2. **Understanding the sequence:** The number of matchsticks increases by the same amount each time. From the given terms: - Pattern 1: 3 matchsticks - Pattern 2: 5 matchsticks - Pattern 3: 7 matchsticks 3. **Finding the term-to-term rule:** Calculate the difference between consecutive terms: $$5 - 3 = 2$$ $$7 - 5 = 2$$ The term-to-term rule is to add 2 matchsticks each time. 4. **Finding the 4th pattern:** Using the rule, the 4th pattern has: $$7 + 2 = 9$$ matchsticks. 5. **Verification by drawing:** The 4th pattern would be four triangles linked in a chain, sharing sides, which totals 9 matchsticks. 6. **Sequence problem 2:** Given sequence: 13, __, 25, ... The terms increase by the same amount each time. Find the missing term. 7. **Calculate the common difference:** Let the missing term be $x$. $$x - 13 = 25 - x$$ $$2x = 38$$ $$x = 19$$ 8. **Sequence problem 3:** Given sequence: 7, __, __, __, 19, ... Find the three missing terms. 9. **Calculate the common difference:** There are 4 intervals between 7 and 19. $$d = \frac{19 - 7}{4} = 3$$ 10. **Find missing terms:** Second term: $$7 + 3 = 10$$ Third term: $$10 + 3 = 13$$ Fourth term: $$13 + 3 = 16$$ 11. **Sequence problem 4:** Given sequence: 250, 50, 10, 2, __, ... Find the next term. 12. **Find the term-to-term rule:** Calculate ratios: $$\frac{50}{250} = 0.2$$ $$\frac{10}{50} = 0.2$$ $$\frac{2}{10} = 0.2$$ The sequence is multiplied by 0.2 each time. 13. **Find the next term:** $$2 \times 0.2 = 0.4$$ **Final answers:** a) Term-to-term rule for matchsticks: add 2 matchsticks. b) Number of matchsticks in 4th pattern: 9. Missing term in sequence 13, __, 25: 19. Missing terms in 7, __, __, __, 19: 10, 13, 16. Next term in 250, 50, 10, 2, __: 0.4.