Question

Consider the sequence(an)n≥1that starts1,3,5,7,9,...(i.e, the odd numbers in order).

(a) Give a recursive definition and closed formula for the sequence.

(b) Write out the sequence(bn)n≥2 of partial sums of (an). Write down the recursive definition for (bn) and guess at the closed formula.

(b) How did you get the partial sums?

Answer #1

Consider the sequence (an)n≥0 which begins 3,8,13,18,23,28,...
(note this means a0 = 3) (a) Find the recursive and closed formulas
for the above sequence. (b) How does the sequence (bn)n≥0 which
begins 3,11,24,42,65,93,... relate to the original sequence (an)?
Explain. (c) Find the closed formula for the sequence (bn) in part
(b) (note, b0 = 3). Show your work.

Provide a recursive definition of some sequence of numbers or
function (e.g. log, exponent, polynomial). Choose one different
from that of any posted thus far. Write a recursive method that
given n, computes the nth term of that sequence.
Also provide an equivalent iterative implementation. How do the two
implementations compare?

We are given a sequence of numbers: 1, 3, 5, 7, 9, . . . and
want to prove that the closed formula for
the sequence is an = 2n – 1.
What would the next number in the sequence be?
What is the recursive formula for the
sequence?
Is the closed formula true for
a1?
What about a2?
What about a3?
Critical Thinking
How many values would we have to check before we could be sure
that the...

Do expand-guess-verify technique for the
following relationships, show step by step.
Find closed-form formula for these recursive relationships.
Estimate running time complexity (Big Oh) of the closed-form
formulas – those also give you idea how “good” or “bad” original
recursive algorithms are!
f(1) = 5
f(n) = f(n-1) + 4
f(1) = 2
f(n) = 3f(n-1)

Consider the problem of summing n numbers by adding together
various pairs of numbers and/or partial sums, for example,
{[(3+1)+(2+5)]+9}.
(a) Represent this addition process with a tree. What will internal
vertices represent?
(b) What is the smallest possible height of an “addition tree” for
summing 100 numbers?

Consider the sequence g0 = 1,
g1 = 1, g2 = 21,
g3 = 41, g4 = 461,
g5 = 1281, g6 = 10501,...
whose linear generator is gn+2 =
gn+1 + 20gn, that is, 20(!)
pairs of baby rabbit offspring.
As we did for the Fibonacci numbers, please derive a closed
form expression for gn.
Consider the sequence hn =
(–1)n gn:
1,–1,21,–41,461,–1281,10501,... Please give a second order
homogeneous linear recurrence with constant coefficients for
hn and prove that...

Ques.3: Consider the addition of n-bit numbers on a machine of
width n-bits. Specifically, the two n-bit 2’s complement numbers
A=an-1,an-2,…a1,a0 and B=bn-1,bn-2,…b-1,b0 are added to get the
n-bit sum C=cn-1,cn-2,…c1,c0. Provide a Boolean function F (defined
in terms of the input variables ai and bi and output variables ci,
where 0 <= i < n ) that determines if an overflow has
occurred in this addition – the function should be expressed in
terms of input variables. F should...

1.
a. Consider the definition of relation. If A is the set of even
numbers and ≡ is the subset of ordered pairs (a,b) where a<b in
the usual sense, is ≡ a relation? Explain.
b. Consider the definition of partition on the
bottom of page 18. Theorem 2 says that the equivalence classes of
an equivalence relation form a partition of the set. Consider the
set ℕ with the equivalence relation ≡ defined by the rule: a≡b in ℕ...

For the sequence 8x + 4, 7x + 3, 6x + 2, 5x +1, ... ,
a. Identify the next 3 terms.
b. Is the sequence arithmetic or geometric? How do you know?
c. Find the explicit and recursive formulae for this
sequence.
d. Write out the sum formula for the first 20 terms and
evaluate.
e. Write your process to part (d) in Sigma Notation.

Consider sequences of n numbers, each in the set {1, 2, . . . ,
6}
(a) How many sequences are there if each number in the sequence
is distinct?
(b) How many sequences are there if no two consecutive numbers
are equal
(c) How many sequences are there if 1 appears exactly i times in
the sequence?

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