Exercise · ZW-HOMOGENEOUS-LINEAR-RECURRENCE-009 · ★★★ Advanced
Solve \(a_n=4a_{n-1}-4a_{n-2}\) with \(a_0=1\) and \(a_1=0\).
Choose one option.
Show answer
- B. \(a_n=(1-n)\,2^n\)
Explanation
The equation \(r^2-4r+4=(r-2)^2=0\) has the double root 2, so \(a_n=(\alpha+\beta n)2^n\). Then \(\alpha=1\) and \(2(1+\beta)=0\), hence \(\beta=-1\).
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.
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