benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / A sequence of functions {f_(n)(x)} is defined recursively as follows: …sqrt {makebox[5mm]{}} is understood to represent the positive square root .) For each positive integer n , …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

A sequence of functions {fn(x)}\, \{f_n(x) \} \, is defined recursively as follows: \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*} (Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root .) For each positive integer nn , find all real solutions of the equation fn(x)=2x\, f_n(x) = 2x \, .
Plain-text mathematical notation (without MathML)
A sequence of functions   {f_(n)(x)}  is defined recursively as follows: \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*} (Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root .) For each positive integer n , find all real solutions of the equation  f_(n)(x)=2x  .
Original LaTeX notation
A sequence of functions  $\, \{f_n(x) \} \,$ is defined recursively as follows: \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*} (Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root .) For each positive integer $n$ , find all real solutions of the equation $\, f_n(x) = 2x \,$ .

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
731581c7-fd1e-596b-a221-54878a813117
task_key
test--731581c7-fd1e-596b-a221-54878a813117
task_revision_id
1
config / split
/ test

Official source

initial import