Kenneth Rosen Edition 7 Exercise 8.2 Question 1 (Page No. 524)

Question

Determine which of these are linear homogeneous recurrence relations with constant coefficients. Also, find the degree of those that are.

• A. $a_{n} = 3a_{n−1} + 4a_{n−2} + 5a_{n−3}$ 
• B. $a_{n} = 2na_{n−1} + a_{n−2}$ 
• C. $a_{n} = a_{n−1} + a_{n−4}$ 
• D. $a_{n} = a_{n−1} + 2 $
• E. $a_{n} = a^{2}_{n−1} + a_{n−2} $
• F. $a_{n} = a_{n−2}$
• G. $a_{n} = a_{n−1} + n$

Answer 1

(a) Given:
$a_{n}=3a_{n-1}+4a_{n-2}+5a_{n-3}$
We note that the given recurrence relation is linear, because the right-hand
side is a linear combination of previous terms.
We note that the given recurrence relation is homogeneous, because the
recurrence relation does not contain any constant terms.
The degree of the recurrence relation is the largest value k for which ark
occurs in the recurrence relation.
k=3

(b) Given:
$a_{n}=2na_{n-1}+a_{n−2}+a_{n-2}$
We note that the given recurrence relation is not linear, because the
right-hand side is not a linear combination of previous terms (due to the
fact that 2n is not a constant).
We note that the given recurrence relation is homogeneous, because the
recurrence relation does not contain any constant terms.

(c) Given:
$a_{n}=a_{n−1}+a_{n−4}$
We note that the given recurrence relation is linear, because the right-hand
side is a linear combination of previous terms.
We note that the given recurrence relation is homogeneous, because the
recurrence relation does not contain any constant terms.
The degree of the recurrence relation is the largest value k for which ar-k
occurs in the recurrence relation.

(d) Given:
$a_{n}=a_{n−1}+2$
We note that the given recurrence relation is linear, because the right-hand
side is a linear combination of previous terms.
We note that the given recurrence relation is not homogeneous, because
the recurrence relation contains a constant term 2.

(e) Given:
$a_{n}=a_{n-1}^{2}+a_{n-2}$
We note that the given recurrence relation is not linear, because the
right-hand side is not a linear combination of previous terms (due to the
fact that is squared).
We note that the given recurrence relation is homogeneous, because the
recurrence relation does not contain any constant terms.
 

(f) Given:
$a_{n}=a_{n−2}$
We note that the given recurrence relation is linear, because the right-hand
side is a linear combination of previous terms.
We note that the given recurrence relation is homogeneous, because the
recurrence relation does not contain any constant terms.
The degree of the recurrence relation is the largest value k for which ar-k
occurs in the recurrence relation.
k=2

(g) Given:
$a_{n}=a_{n−1}+n$
We note that the given recurrence relation is not linear, because the
right-hand side is not a linear combination of previous terms (due to the
fact that n is not a constant or a previous term).
We note that the given recurrence relation is not homogeneous, because
n is not a multiply of a term an

 

(a) Linear, homogeneous, degree 3
(b) Not linear, homogeneous
(c) Linear, homogeneous, degree 4
(d) Linear, not homogeneous
(e) Not linear, homogeneous
(f) Linear, homogeneous, degree 2
(g) Not linear, not homogeneous