Sequences & Series
Iterative averaging sequence converging to √5
MJAT_TS6_P1
Grade 12
Question:
Let $x$ be a positive number. A sequence $\{x_n\}$ is defined by $x_1=\frac{1}{2}\left(x+\frac{5}{x}\right)$, $x_2=\frac{1}{2}\left(x_1+\frac{5}{x_1}\right)$, ..., $x_{n+1}=\frac{1}{2}\left(x_n+\frac{5}{x_n}\right)$. Then which of the following is/are true?
A) For all $n\geq 1$: $\dfrac{x_n-\sqrt{5}}{x_n+\sqrt{5}}=\left(\dfrac{x-2\sqrt{5}}{x+5}\right)^{2^n}$... (some form)
B) $\displaystyle\lim_{n\to\infty}x_n=\sqrt{5}$
C) For all $n\geq 1$: $\dfrac{x_n-\sqrt{5}}{x_n+\sqrt{5}}=\left(\dfrac{x+\sqrt{5}-2\cdot 5}{x}\right)^{2^n}$
D) $\displaystyle\lim_{n\to\infty}x_n=\frac{1}{\sqrt{5}}$
Step-by-Step Solution
Key Concept: This is Newton's method (Babylonian method) for $\sqrt{5}$. Define $r_n=\frac{x_n-\sqrt{5}}{x_n+\sqrt{5}}$. Then $r_{n+1}=r_n^2$ (squaring property). So $r_n=r_0^{2^n}$.
A ✓, B ✓. Answer: A, B.
Correct Answer: AB