Question

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

Answer #1

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

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)

Higher order recurrence relations:
Find the general solution to each recurrence
(a) an −6an−1 + 11an−2 −6an−3 = 0 for n ≥ 3.
(b) an −5an−2 + 4an−4 = 0 for n ≥ 4.
*hint* To factor a polynomial of degree 3 or more
look for a root r = a then use long division (dividing by the
linear factor r −a) to ﬁnd the cofactor (which will now be of
degree one less). Repeat until you have factored completely into...

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

Consider the recurrence relation T(1) = 0, T(n) = 25T(n/5) +
5n.
(a) Use the Master Theorem to find the order of magnitude of
T(n)
(b) Use any of the various tools from class to find a closed-form
formula for T(n), i.e. exactly solve the recurrence.
(c) Verify your solution for n = 5 and n = 25.

We denote {0, 1}n by sequences of 0’s and 1’s of
length n. Show that it is possible to order elements of {0,
1}n so that two consecutive strings are different only
in one position

Find the d.f.t of two finite sequences f[n]= -3,-1,-7 and g[n]=
4, 0, 1/2. and find the convolution.
thank you

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

Here are two relations:
R(A,B): {(0, 1), (2,3), (0, 1), (2,4), (3,4)}
S(B, C): {(0, 1), (2, 4), (2, 5), (3, 4), (0, 2), (3, 4)}
Compute the following: a) 11'A+B.A2,B2(R); b) 71'B+l,C-l(S); c)
TB,A(R); d) TB,c(S); e) J(R); f) J(S); g) /A, SUM(Bj(R); h)
IB.AVG(C)(S'); ! i) !A(R); ! j) IA,MAX(C)(R t:><1 S); k) R ~L
S; I) R ~H S; m) R ~ S; n) R ~R.B<S.B S.
I want to know the solution for j to m

1) State the main difference between an ODE and a PDE?
2) Name two of the three archetypal PDEs?
3) Write the equation used to compute the Wronskian for two
differentiable
functions, y1 and y2.
4) What can you conclude about two differentiable functions, y1 and
y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what
does the Principle of
Superposition guarantee?
6) (8 pts, 4 pts each) State...

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