A second-order linear homogeneous recurrence relation with constant coefficients is a recurrence relation of the form _____ for all integers k _____, where _____.

Given a recurrence relation of the form a_k= A a_(k-1)+ B a_(k-2) for all integers k 2, the characteristic equation of the relation is _____.

If a sequence a_1,a_2,a_3, is defined by a second-order linear homogeneous recurrence relation with constant coefficients and the characteristic equation for the relation has two distinct roots r and s (which could be complex numbers), then the sequence is given by an explicit formula of the form _____.

If a sequence a_1,a_2,a_3, is defined by a second-order linear homogeneous recurrence relation with constant coefficients and the characteristic equation for the relation has only a single root r , then the sequence is given by an explicit formula of the form _____.

Problem 2

Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence.

a_k=7a_(k-1)-10a_(k-2) for all integers k2

a_0=2,a_1=2

Problem 3

Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence.

a_k=a_(k-1)+6a_(k-2) for all integers k2

a_0=0,a_1=3

Problem 4

Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence.

a_k=4a_(k-2) for all integers k2

a_0=1,a_1=-1

Problem 5

Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence.

r_k=2r_(k-1)-r_(k-2) for all integers k2

r_0=1,r_1=4

Problem 6

Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence.

r_k=4r_(k-1)-4r_(k-2) for all integers k2

r_0=0,r_1=3

Problem 7

Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence.

r_k=2r_(k-1)+2 r_(k-2) for all integers k2

r_0=1,r_1=3