QuestionFind the fixed points of the sequence defined by $a_{n+1}=\sqrt{30 a_{n}}$ .

Studdy Solution

STEP 1

Assumptions1. The sequence $\left\{a_{n}\right\}$ is recursively defined by the equation $a_{n+1}=\sqrt{30 a_{n}}$ . . A fixed point of a sequence is a value $a$ such that $f(a) = a$ .

STEP 2

To find the fixed points, we need to set $a_{n+1} = a_{n}$ in the recursive equation and solve for $a_{n}$ .
$a_{n} = \sqrt{30 a_{n}}$

STEP 3

Square both sides of the equation to eliminate the square root.
$(a_{n})^2 =30 a_{n}$

STEP 4

Rearrange the equation to a standard quadratic form.
$(a_{n})^2 -30 a_{n} =0$

STEP 5

Factor out $a_{n}$ from the equation.
$a_{n}(a_{n} -30) =0$

STEP 6

Set each factor equal to zero and solve for $a_{n}$ .
$a_{n} =0$ $a_{n} -30 =0 \Rightarrow a_{n} =30$ The fixed points of the sequence $\left\{a_{n}\right\}$ are0 and30.

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QuestionFind the fixed points of the sequence defined by $a_{n+1}=\sqrt{30 a_{n}}$ .

Studdy Solution

STEP 1

Assumptions1. The sequence $\left\{a_{n}\right\}$ is recursively defined by the equation $a_{n+1}=\sqrt{30 a_{n}}$ . . A fixed point of a sequence is a value $a$ such that $f(a) = a$ .

STEP 2

To find the fixed points, we need to set $a_{n+1} = a_{n}$ in the recursive equation and solve for $a_{n}$ .
$a_{n} = \sqrt{30 a_{n}}$

STEP 3

Square both sides of the equation to eliminate the square root.
$(a_{n})^2 =30 a_{n}$

STEP 4

Rearrange the equation to a standard quadratic form.
$(a_{n})^2 -30 a_{n} =0$

STEP 5

Factor out $a_{n}$ from the equation.
$a_{n}(a_{n} -30) =0$

STEP 6

Set each factor equal to zero and solve for $a_{n}$ .
$a_{n} =0$ $a_{n} -30 =0 \Rightarrow a_{n} =30$ The fixed points of the sequence $\left\{a_{n}\right\}$ are0 and30.

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QuestionFind the fixed points of the sequence defined by an+1=30ana_{n+1}=\sqrt{30 a_{n}}an+1​=30an​​.

Studdy Solution

STEP 1

Assumptions1. The sequence {an}\left\{a_{n}\right\}{an​} is recursively defined by the equation an+1=30ana_{n+1}=\sqrt{30 a_{n}}an+1​=30an​​. . A fixed point of a sequence is a value aaa such that f(a)=af(a) = af(a)=a.

STEP 2

To find the fixed points, we need to set an+1=ana_{n+1} = a_{n}an+1​=an​ in the recursive equation and solve for ana_{n}an​.an=30ana_{n} = \sqrt{30 a_{n}}an​=30an​​

STEP 3

Square both sides of the equation to eliminate the square root.(an)2=30an(a_{n})^2 =30 a_{n}(an​)2=30an​

STEP 4

Rearrange the equation to a standard quadratic form.(an)2−30an=0(a_{n})^2 -30 a_{n} =0(an​)2−30an​=0

STEP 5

Factor out ana_{n}an​ from the equation.an(an−30)=0a_{n}(a_{n} -30) =0an​(an​−30)=0

STEP 6

Set each factor equal to zero and solve for ana_{n}an​.an=0a_{n} =0an​=0an−30=0⇒an=30a_{n} -30 =0 \Rightarrow a_{n} =30an​−30=0⇒an​=30The fixed points of the sequence {an}\left\{a_{n}\right\}{an​} are0 and30.

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QuestionFind the fixed points of the sequence defined by $a_{n+1}=\sqrt{30 a_{n}}$ .

Assumptions1. The sequence $\left\{a_{n}\right\}$ is recursively defined by the equation $a_{n+1}=\sqrt{30 a_{n}}$ . . A fixed point of a sequence is a value $a$ such that $f(a) = a$ .

To find the fixed points, we need to set $a_{n+1} = a_{n}$ in the recursive equation and solve for $a_{n}$ .
$a_{n} = \sqrt{30 a_{n}}$

Square both sides of the equation to eliminate the square root.
$(a_{n})^2 =30 a_{n}$

Rearrange the equation to a standard quadratic form.
$(a_{n})^2 -30 a_{n} =0$

Factor out $a_{n}$ from the equation.
$a_{n}(a_{n} -30) =0$

Set each factor equal to zero and solve for $a_{n}$ .
$a_{n} =0$ $a_{n} -30 =0 \Rightarrow a_{n} =30$ The fixed points of the sequence $\left\{a_{n}\right\}$ are0 and30.