Question

Recurrence Relations

Solve the following recurrence equation:

f(n, k) = 0, if k > n

f(n,k) = 1, if k = 0

f(n,k) = f(n-1, k) + f(n-1,k-1), if n >= k > 0

Answer #1

Two inﬁnite sequences {an}∞ n=0 and {bn}∞
n=0 satisfy the recurrence relations an+1 =
an −bn and bn+1 = 3an +
5bn for all n ≥ 0. Imitate the techniques
used to solve diﬀerential equations to ﬁnd general formulas for
an and bn in terms of n.

Solve the following recurrence relations:
1.
T(n) = T(n/3) + n for n > 1
T(1) = 1
2.
T(n) = 4T(n/2) + n^2 for n > 1
T(1) = 1

Solve the following sets of recurrence relations and initial
conditions:
S(k)−2S(k−1)+S(k−2)=2, S(0)=25, S(1)=16
The answer is: S(k)=k^2−10k+25
Please help me understand the solution. I get how to get the
homogenous solution, I would get Sh(k) = a + bk
But I get stuck on the particular solution. Thanks

Also write the time complexity
Solve the non-linear recurrence equation using recurrence
A(n) = 2A(n/2) + n
Solve the non-linear recurrence equation using Master’s
theorem
T (n) = 16T (n/4) + n

Solve the following recurrence relations. If possible, use the
Master Theorem. If the Master Theorem is not possible, explain why
not and solve it using another approach (substitution with n =
3h or h - log3(n).
a. T(n) = 7 * n2 + 11 * T(n/3)
b. T(n) = 4 * n3 * log(n) + 27 * T(n/3)

Problem #1.
Solve the following recurrence exactly.
9n^2 - 15n +
106
if n = 0, 1 or 2
t(n)=
t(n-1) + 2t(n-2) -
2t(n-3)
otherwise
Problem #2.
Solve the following recurrence exactly.
n
if n = 0, 1 2, or 3
t(n)=
t(n-1) + t(n-3) -
t(n-4)
otherwise
Problem #3.
Solve the following recurrence exactly.
n +
1
if n = 0, or 1
t(n)=
3t(n-1) - 2t(n-2) +...

Solve f(n) as a function of n using the homogeneous equation and
given the conditions below:
f(0) = 0;
f(1) = 1;
f(2) = 4;
f(n) = 2 f(n-1) - f(n-2) + 2; n >
2

2. Find a system of recurrence relations for the number of
n-digit quaternary sequences that contain an even number of 2’s and
an odd number of 3’s. Define the initial conditions for the system.
(A quaternary digit is either a 0, 1, 2 or 3)

Find a closed form for the following recurrence relations. Show
your work. (a) an = −an−1, a0 = 3 (b) an = an−1 − n, a0 = 5 (c) an
= 2an−1 − 3, a0 = 2

Solve the following recurrence relation
T(1) = c1
T(n) = 2*T(n/2) + c2

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