Exercise · ZW-HOMOGENEOUS-LINEAR-RECURRENCE-010 · ★★★ Advanced
Which statement about solutions of a homogeneous linear recurrence is true?
Choose one option.
Show answer
- D. Every linear combination of solutions is a solution
Explanation
If \(x_n\) and \(y_n\) satisfy the recurrence, so does \(\alpha x_n+\beta y_n\), by linearity. Products and adding a constant do not preserve it, and without initial values there are infinitely many solutions.
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-004 What is the characteristic equation of the Fibonacci sequence a_n=a_n-1+a_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.