a) let t_(1)=2,t_(2)=4,t_(3)=8 and forgeqslant 1,t_(n+3)=t_(n+2)+t_(n+1)+2t_(n) Find a pattern for t_(n) and prove your answer. (4 marks) b) Prove that for any integer x, x is odd if and only if x^2 is odd. (5 marks) c) Prove by contradiction that the sum of a rational number and an irrational number in irrational (6 marks) d) Prove that sqrt (2)+1 is an irrational number (5 marks)

a) To find a pattern for , we can start by calculating the first few terms of the sequence: Now, let's use the given recursive formula to calculate the next terms: From these calculations, we can observe that .To prove this pattern, we can use mathematical induction.Base case: For , we have , which is true.Inductive step: Assume that the pattern holds for some . We need to show that it holds for .Using the recursive formula, we have: Substituting the pattern for , , and , we get: Simplifying the expression, we have: Combining like terms, we get: Therefore, the pattern holds for , and by the principle of mathematical induction, the pattern holds for all .b) To prove that is odd if and only if is odd, we can use a direct proof.Suppose is odd. Then, we can write for some integer . Squaring both sides, we get: Since is an integer, is odd.Now, suppose is odd. Then, we can write for some integer . Taking the square root of both sides, we get: Since is odd, must be odd.Therefore, is odd if and only if is odd.c) To prove by contradiction that the sum of a rational number and an irrational number is irrational, we can assume the opposite, i.e., that the sum of a rational number and an irrational number is rational.Let be a rational number and be an irrational number. Suppose their sum is rational. Then, we can write for some rational number $