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MATH / The Fibonacci numbers are defined recursively by the equation

Problem

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problem

The Fibonacci numbers are defined recursively by the equation Fn=Fn−1+Fn−2 F_n = F_{n - 1} + F_{n - 2}for every integer n≥2 n \ge 2, with initial values F0=0 F_0 = 0 and F1=1 F_1 = 1. Let Gn=F3n G_n = F_{3n} be every third Fibonacci number. There are constants a a and b b such that every integer n≥2 n \ge 2 satisfies Gn=aGn−1+bGn−2. G_n = a G_{n - 1} + b G_{n - 2}.Find (a,b)(a,b).
Plain-text mathematical notation (without MathML)
The Fibonacci numbers are defined recursively by the equation
F_(n)=F_(n−1)+F_(n−2)for every integer n≥2, with initial values F₀=0 and F₁=1.  Let G_(n)=F_(3n) be every third Fibonacci number.  There are constants a and b such that every integer n≥2 satisfies
G_(n)=aG_(n−1)+bG_(n−2).Find (a,b).
Original LaTeX notation
The Fibonacci numbers are defined recursively by the equation
\[ F_n = F_{n - 1} + F_{n - 2}\]for every integer $ n \ge 2$, with initial values $ F_0 = 0$ and $ F_1 = 1$.  Let $ G_n = F_{3n}$ be every third Fibonacci number.  There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies
\[ G_n = a G_{n - 1} + b G_{n - 2}.\]Find $(a,b)$.

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
2f0ff523-c3c2-5c66-92d8-45efb64f094b
task_key
intermediate~5falgebra--test--2f0ff523-c3c2-5c66-92d8-45efb64f094b
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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