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Please answer questions 2, 3 and 4, thank you

Please answer questions 2, 3 and 4, thank you

Please answer questions 2, 3 and 4, thank you

For questions 1 and 2 , make sure you set out and explain your proofs clearly. To receive marks for question 3, your answers must be exactly right and use correct notation. (2) Let \( a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of integers defined by \( a_{1}=2, a_{2}=8, a_{3}=25, \quad \) and \( \quad a_{i}=2 a_{i-1}+2 a_{i-2}+a_{i-3} \) for each integer \( i \geq 4 \). Prove by strong induction that \( a_{n}<3^{n} \) for all integers \( n \geq 1 \). (3) Let \( R, S \) and \( T \) be sets defined as follows. \[ \begin{array}{l} R=\mathbb{N}-\{5,6,7\} \\ S=\{x \in \mathbb{Z}: 1 \leq x \leq 8\} \\ T=\{x \in \mathbb{Z}: x \geq 7\} \end{array} \] Find the following. (i) \( R \cap S \) (ii) \( R-T \) (iii) \( (S \cap T) \times(S-R) \) (iv) \( \{\{\},\{3\},\{4,8,9\},\{8\},\{9,10\},\{x \in \mathbb{Z}: x \geq 11\}\}-\mathcal{P}(T) \) (v) \( |\mathcal{P}(T-R) \times S| \) [No explanation required.] (4) Which of the following is true? (a) \( (A \times C) \cup(B \times C)=(A \cup B) \times C \) for all sets \( A, B \) and \( C \). (b) \( (A \times C) \cup(B \times C) \neq(A \cup B) \times C \) for all sets \( A, B \) and \( C \). (c) \( (A \times C) \cup(B \times C)=(A \cup B) \times C \) for some sets \( A, B \) and \( C \), but \( (A \times C) \cup(B \times C) \neq(A \cup B) \times C \) for some other sets sets \( A, B \) and \( C \).