QuestionFind fixed points of the sequence defined by $a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{3}{a_{n}}\right)$ .

Studdy Solution

STEP 1

Assumptions1. We are looking for fixed points of the sequence $\left\{a_{n}\right\}$ . . A fixed point of a sequence is a value $a$ such that $a_{n+1} = a_n = a$ for some $n$ .
3. The sequence is defined by the recursive formula $a_{n+1}=\frac{1}{}\left(a_{n}+\frac{3}{a_{n}}\right)$ .

STEP 2

To find the fixed points, we need to solve the equation $a_{n+1} = a_n = a$ for $a$ . This gives us the equation $a = \frac{1}{2}\left(a + \frac{}{a}\right)$

STEP 3

Multiply both sides of the equation by $2a$ to clear the fraction.
$2a^2 = a^2 +3$

STEP 4

Subtract $a^2$ from both sides of the equation to isolate the terms involving $a$ on one side.
$a^2 =3$

STEP 5

Take the square root of both sides of the equation to solve for $a$ . Remember that the square root has two solutions, one positive and one negative.
$a = \sqrt{3},\, a = -\sqrt{3}$ The fixed points of the sequence are $\sqrt{3}$ and $-\sqrt{3}$ .

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