1. **Problem Statement:** We have several sequences and factorization problems to solve. 2. **Sequence 1:** 250, 50, 10, 2, ... - Identify the term-to-term rule. - Find the next term. 3. **Sequence 2:** 8, 14, 20, 26, ... - Find the nth term rule for this arithmetic sequence. 4. **Prime Factor Trees:** - For 30 and 42, find common prime factors and HCF. - For 66 and 110, find the HCF. --- ### Sequence 1: 250, 50, 10, 2, ... - Term-to-term rule: Each term is divided by 5. - Check: $\frac{250}{5} = 50$, $\frac{50}{5} = 10$, $\frac{10}{5} = 2$ - Next term: $\frac{2}{5} = 0.4$ ### Sequence 2: 8, 14, 20, 26, ... - This is an arithmetic sequence with common difference $d = 14 - 8 = 6$ - First term $a_1 = 8$ - Formula for nth term of arithmetic sequence: $$a_n = a_1 + (n-1)d$$ - Substitute values: $$a_n = 8 + (n-1)6 = 8 + 6n - 6 = 6n + 2$$ ### Prime Factor Trees for 30 and 42: - Prime factors of 30: $2, 3, 5$ - Prime factors of 42: $2, 3, 7$ **a)** Common prime numbers: $2$ and $3$ **b)** HCF as product of common prime factors: $$2 \times 3 = 6$$ **c)** HCF of 30 and 42 is $6$ ### HCF of 66 and 110: - Prime factors of 66: $2, 3, 11$ - Prime factors of 110: $2, 5, 11$ - Common prime factors: $2$ and $11$ - HCF: $$2 \times 11 = 22$$ --- **Final answers:** - Next term in sequence 1: $0.4$ - nth term of sequence 2: $a_n = 6n + 2$ - HCF of 30 and 42: $6$ - HCF of 66 and 110: $22$