Question: A sequence starts at $10$. The term-to-term rule is to subtract $3$ then multiply by $2$ each time. a) Work out the fourth term of the sequence. b) Is this sequence increasing, decreasing or neither?

1. **State the problem:** We have a sequence starting at $10$. The rule to get from one term to the next is: subtract $3$ then multiply by $2$. We want to find the fourth term and determine if the sequence is increasing, decreasing, or neither. 2. **Write the rule mathematically:** If $a_n$ is the $n$th term, then $$a_{n+1} = (a_n - 3) \times 2$$ with $a_1 = 10$. 3. **Calculate terms step-by-step:** - First term: $a_1 = 10$ - Second term: $$a_2 = (a_1 - 3) \times 2 = (10 - 3) \times 2 = 7 \times 2 = 14$$ - Third term: $$a_3 = (a_2 - 3) \times 2 = (14 - 3) \times 2 = 11 \times 2 = 22$$ - Fourth term: $$a_4 = (a_3 - 3) \times 2 = (22 - 3) \times 2 = 19 \times 2 = 38$$ 4. **Answer to part (a):** The fourth term is $38$. 5. **Determine if the sequence is increasing or decreasing:** Look at the terms: $10, 14, 22, 38$ Each term is larger than the previous one. Therefore, the sequence is **increasing**. **Final answers:** - a) The fourth term is $38$. - b) The sequence is increasing.