Number Theory / Sets
Sets involving a+b√2 and powers of (1±√2)
MJAT_TS7_P1
Grade 12
Question:
Let $S=\{a+b\sqrt{2}: a,b\in\mathbb{Z}\}$, $T_1=\{(-1+\sqrt{2})^n: n\in\mathbb{Z}\}$, $T_2=\{(1+\sqrt{2})^n: n\in\mathbb{Z}\}$. Which of the following is/are TRUE?
A) $\mathbb{Z}\cup T_1\cup T_2\subset S$
B) $T_1\cap\left(0,\dfrac{1}{2024}\right)=\emptyset$
C) $T_2\cap(2024,\infty)\neq\emptyset$
D) For any $a,b\in\mathbb{Z}$: $\cos(\pi(a+b\sqrt{2}))+i\sin(\pi(a+b\sqrt{2}))\in\mathbb{Z}$ iff $b=0$
Step-by-Step Solution
Key Concept: A: $(\pm 1+\sqrt{2})^n=a_n+b_n\sqrt{2}$ for integers $a_n,b_n$ (since Binomial expansion). B: $(-1+\sqrt{2})>0$ and $(-1+\sqrt{2})^n\to 0$ for $n\to+\infty$: for large $n$, elements of $T_1$ fall in $(0,1/2024)$ — B is FALSE. C: $(1+\sqrt{2})^n\to\infty$ for $n\to+\infty$ — some element $>2024$: C is TRUE. D: From Euler's formula.
A ✓, B ✗, C ✓, D ✓. Answer: A, C, D.
Correct Answer: ACD