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ve203 Assignment 8

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ve203 Assignment 8

Q1. Prove that if (an) is a sequence that satisfies a linear homogeneous recurrence
relation of degree 2 whose characteristic polynomial has only one real root α, then there
exists q1, q2 ∈ R such that for all n ∈ N,
an = q1α
n + q2nαn
(4 marks)
Q2. Find an expression for the terms of the sequence (an) that satisfy
an = 2an−1 + an−2 − 2an−3
with a0 = 3, a1 = 6 and a2 = 0.
(2 marks)

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ve203 Assignment 8

Q1. Prove that if (an) is a sequence that satisfies a linear homogeneous recurrence
relation of degree 2 whose characteristic polynomial has only one real root α, then there
exists q1, q2 ∈ R such that for all n ∈ N,
an = q1α
n + q2nαn
(4 marks)
Q2. Find an expression for the terms of the sequence (an) that satisfy
an = 2an−1 + an−2 − 2an−3
with a0 = 3, a1 = 6 and a2 = 0.
(2 marks)
Q3. Find an expression for the terms of the sequence (an) that satisfy
an = 5an−1 − 6an−2 + 2n + 2n
2 + n
with a0 = 0, a1 = 4.
(3 marks)
Q4. Find an expression for the terms of the sequence (an) that satisfy
an = 7an−1 − 16an−2 + 12an−3 + n4
n
with a0 = −3, a1 = 2, a2 = 5.
(2 marks)
Q5. For all n ∈ N\{0}, let
an =
Xn
i=1
i
4
By finding a recurrence relation that (an) satisfies and solving that recurrence relation,
find an expression for the terms of the sequence (an).
(2 marks)
Q6. Solve the simultaneous recurrence relations
an = 3an−1 + 2bn−1
bn = an−1 + 2bn−1
with a0 = 1 and b0 = 2.
(3 marks)
Q7. Find an expression for the terms of the sequence (an) that satisfy
an = 2an−1 − 2an−2 + 3n
with a0 = 1, a1 = 2.
(2 marks)
1
Q8. Find a closed formula for the sequences generated by the following generating
functions:
(i) G(x) = x − 1 + 1
1−3x
(ii) G(x) = e
3x
2
− 1
(iii) G(x) = x
1+x+x2
(3 marks)
Q9.
(i) Consider a recurrence relation in the form
f(n)an = g(n)an−1 + h(n)
for n ≥ 1, and with a0 = C. Show that this recurrence relation can be reduced to
a recurrence relation of the form
bn = bn−1 + Q(n)h(n)
where
bn = g(n + 1)Q(n + 1)an and Q(n) = f(1)f(2)· · · f(n − 1)
g(1)g(2)· · · g(n)
(ii) Use (i) to solve the original recurrence relation and obtain
an =
C +
Pn
i=1 Q(i)h(i)
g(n + 1)Q(n + 1)
(iii) Find an expression for the sequence (an) that satisfies
an = (n + 3)an−1 + n with a0 = 1
(6 marks)
2