1- Show that (n^3+3n^2+3n+1) / (n+1) is O (n2 ). Use the
definition and proof of big-O notation.
2- Prove using the definition of Omega notation that either 8 n
is Ω (5 n ) or not.
please help be with both
3. To begin a proof by contradiction for “If n is even then n+1
is odd,” what would you “assume true?
4. Prove that the following is not true by
finding a counterexample.
“The sum of any 3 consecutive integers is
even"
5. Show a Proof by exhaustion for the
following: For n = 2, 4, 6, n²-1 is
odd
6. Show an informal
Direct Proof for “The sum of 2 even
integers is even.”
Recursive Definitions
7. The Fibonacci Sequence
is...
3. The Hofstadter Conway sequence is defined by a(1)=a(2)=1 and (for n>2 by) a(n)=a(a(n-1))+a(n-a(n-1)). Write a function to quickly compute this sequence.
>>> [hc(i) for i in range(1,20)]
[1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11]
Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for
every n ∈ N. That is, the sum of the first n perfect cubes is the
square of the sum of the first n natural numbers. (As a student, I
found it very surprising that the sum of the first n perfect cubes
was always a perfect square at all.)
Data Structure:
1. Write a program for f(n) = 1^2+2^3+…+n^2. (i^2 = i*i)
2. If you have the following polynomial function
f(n)=a0 +a1 x + a2x2+…+an xn ,
then you are asked to write a program for that, how do you
do?
3. Write a function in C++ to sort array A[]. (You can assume
that you have 10 elements in the array.)
4. Analyze the following program, tell us what does it do for
each location of “???” (...
(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an
epsilon, N argument to show that this is indeed the correct
limit.
(b) Use an epsilon, N argument to show that {1/(n^(1/2))}
converges to 0.
(c) Let k be a positive integer. Use an epsilon, N argument to
show that {a/(n^(1/k))} converges to 0.
(d) Show that if {Xn} converges to x, then the sequence {Xn^3}
converges to x^3. This has to be an epsilon, N argument [Hint: Use
the difference...
Proof of the smoothness rule by providing details of the proof
that t (n) ∈ Ω(f(n)) whenever (n) is a smooth function and t(n) is
an eventually nondecreasing function such that
t(n) ∈ Θ(f(n) | n is a power of b).