Question

consider 1/n. Find all the values of n for which 1/n has a decimal representation with period 1. You may use the fact that numbers of the form n=2^a5^b have a period of 1 without having to prove it.

Answer #1

consider
1/n. Find all the values of n for which 1/n has a decimal
representation with period 1. You may use the fact that numbers of
the form n=2^a5^b have a period of 1 without having to prove
it

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Question 1: A. Convert the following numbers to their decimal
representation. Show your work. 1. 100110112 =
2. 4567 =
3. 38A16
= 4. 22145 = B. Convert the following numbers to their binary
representation: 1. 6910 =
2. 48510=
3. 6D1A16 = C. Convert the
following numbers to their hexadecimal representation: 1. 11010112
=
2. 89510 = Question 2: Solve the following, do all calculation
in the given base. Show your work.

Find all natural numbers n so that
n3 + (n + 1)3 > (n +
2)3.
Prove your result using induction.

Prove that for a square n ×n matrix A, Ax = b (1) has one and
only one solution if and only if A is invertible; i.e., that there
exists a matrix n ×n matrix B such that AB = I = B A.
NOTE 01: The statement or theorem is of the form P iff Q, where
P is the statement “Equation (1) has a unique solution” and Q is
the statement “The matrix A is invertible”. This means...

1. Find the radius of convergence for:
∞∑n=1 (n!)^2 x^n /(2n)!
2. Find all the values of x such that the given
series would converge.
∞∑n=1 (−1)^n x^n / 2^n(n^2+9)
The series is convergent
from x=
to x=

Assume a computer is using signed magnitude base 2
representation to store 8-bit values. Further, assume you have a
program which can display the contents of 1-byte memory locations.
However, the program displays hex (as a shorthand) instead of
binary. What decimal values are represented by the following:
(a) C3
(b) 6E
Repeat the above, but assume the numbers are stored as 8-bit 2’s
complement base 2.

Consider below recurrence relation
f_(n )=f_(n-1)+ f_(n-2) for n ≥ 3. f_(1 )=1 and f_2 = 3
(a) Please compute the first seven numbers in this sequence.
(b) Find the closed form for this recurrence relation. Solving
the characteristic equation, and solving for constants

4. Consider a continuous random variable X which has pdf fX(x) =
1/7, 0 < x < 7.
(a) Find the values of µ and σ^ 2 . (You may recognize the model
above, and if you do, it is OK to simply write down the answers if
you know them.)
(b) A random sample of size n = 28 is taken from the above
distribution. Find, approximately, IP(3.3 ≤ X ≤ 3.51). Hint: use
the CLT.

2.) Consider an input array A of size n in which n − 1 of the
elements have identical values and the remaining one value is
smaller than the n − 1 identical values. What is the running time
of Heapsort with input A?

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