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i need some one to help me solve the questions in the document .
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1. A second-order linear homogeneous recurrence relation with constant coefficients
is a recurrence relation of the form __
___ for all integers k ≥ _____,
where __
___.
2. Given a recurrence relation of the form 𝑎𝑘 = 𝐴 𝑎𝑘−1 + 𝐵 𝑎𝑘−2 for all
integers 𝑘 ≥ 2 , the characteristic equation of the relation is ___
_
_.
3. If a sequence 𝑎1 , 𝑎2 , 𝑎3 , ⋯ 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 __
_
_
_.
4. If a sequence 𝑎1 , 𝑎2 , 𝑎3 , ⋯ 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.
𝑎𝑘 = 7𝑎𝑘−1 − 10𝑎𝑘−2 for all integers k ≥ 2
𝑎0 = 2, 𝑎1 = 2
Problem 3
Suppose a sequence satisfies the below given recurrence relation and initial conditions.
Find an explicit formula for the sequence.
𝑎𝑘 = 𝑎𝑘−1 + 6𝑎𝑘−2 for all integers k ≥ 2
𝑎0 = 0, 𝑎1 = 3
Problem 4
Suppose a sequence satisfies the below given recurrence relation and initial conditions.
Find an explicit formula for the sequence.
𝑎𝑘 = 4𝑎𝑘−2 for all integers k ≥ 2
𝑎0 = 1, 𝑎1 = −1
Problem 5
Suppose a sequence satisfies the below given recurrence relation and initial conditions.
Find an explicit formula for the sequence.
𝑟𝑘 = 2𝑟𝑘−1 − 𝑟𝑘−2 for all integers k ≥ 2
𝑟0 = 1, 𝑟1 = 4
Problem 6
Suppose a sequence satisfies the below given recurrence relation and initial conditions.
Find an explicit formula for the sequence.
𝑟𝑘 = 4𝑟𝑘−1 − 4𝑟𝑘−2 for all integers k ≥ 2
𝑟0 = 0, 𝑟1 = 3
Problem 7
Suppose a sequence satisfies the below given recurrence relation and initial conditions.
Find an explicit formula for the sequence.
𝑟𝑘 = 2𝑟𝑘−1 + 2 𝑟𝑘−2 for all integers k ≥ 2
𝑟0 = 1, 𝑟1 = 3
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