Exercise · ZW-HOMOGENEOUS-LINEAR-RECURRENCE-004 · ★☆☆ Basic
What is the characteristic equation of the Fibonacci sequence \(a_n=a_{n-1}+a_{n-2}\)?
Choose one option.
Show answer
- D. \(r^2-r-1=0\)
Explanation
Substituting \(a_n=r^n\) and dividing by \(r^{n-2}\) gives \(r^2=r+1\), that is \(r^2-r-1=0\), with roots \(\tfrac{1\pm\sqrt5}{2}\).
Review the concept: Homogeneous linear recurrence
A homogeneous linear recurrence of order \(k\) expresses each term \(a_n\) as a combination of the previous \(k\) terms with constant coefficients \(c_1,\dots,c_k\), with no independent term.
\[a_n=c_1a_{n-1}+c_2a_{n-2}+\cdots+c_ka_{n-k},\quad c_k\neq0\]
More practice problems: homogeneous linear recurrence
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-001 Which of these is a homogeneous linear recurrence with constant coefficients?
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-002 What is the order of a_n=5a_n-1-6a_n-3?
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-003 What are the roots of the characteristic equation of a_n=5a_n-1-6a_n-2?
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-005 If a_n=2a_n-1+3a_n-2, a_0=1 and a_1=2, what is a_5?
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-006 If a_n=-a_n-2, a_0=1 and a_1=0, what is a_10?
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-007 How many initial values determine a unique solution of a homogeneous linear recurrence of order 4?
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-008 Solve a_n=5a_n-1-6a_n-2 with a_0=1 and a_1=5.
- ZW-HOMOGENEOUS-LINEAR-RECURRENCE-009 Solve a_n=4a_n-1-4a_n-2 with a_0=1 and a_1=0.
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