Recall the Fibonacci sequence: 1,1,2,3,8,13....
In general we can generate Fibonacci-like sequences by making
linear combinations...
Recall the Fibonacci sequence: 1,1,2,3,8,13....
In general we can generate Fibonacci-like sequences by making
linear combinations of the formula that gives us the nth term of
the sequence.Consider the following general case for generating
Fibonacci-like sequences:
F 1 = 1
F 2 = 1
Fn = aFn-1 + bFn-2
where a, b are integers .
Write a function that given values a, b and n will print a
Fibonacci-like sequence up to the nth term of the sequence.
( eg, if...
) Let α be a fixed positive real number, α > 0. For a
sequence {xn},...
) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...
Consider the sequence (xn)n given by x1 = 2, x2 = 2 and xn+1 =
2(xn...
Consider the sequence (xn)n given by x1 = 2, x2 = 2 and xn+1 =
2(xn + xn−1).
(a) Let u, w be the solutions of the equation x 2 −2x−2 = 0, so
that x 2 −2x−2 = (x−u)(x−w). Show that u + w = 2 and uw = −2.
(b) Possibly using (a) to aid your calculations, show that xn =
u^n + w^n .
(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then...
(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}
2. Let f(x) = sin(2x) and x0 = 0.
(A) Calculate the Taylor approximation T3(x)
(B)....
2. Let f(x) = sin(2x) and x0 = 0.
(A) Calculate the Taylor approximation T3(x)
(B). Use the Taylor theorem to show that
|sin(2x) − T3(x)| ≤ (2/3)(x − x0)^(4).
(C). Write a Matlab program to compute the errors for x = 1/2^(k)
for k = 1, 2, 3, 4, 5, 6, and verify that
|sin(2x) − T3(x)| = O(|x − x0|^(4)).
2) Let a, b and c be any
integers that form a perfect triangle, i.e. satisfy...
2) Let a, b and c be any
integers that form a perfect triangle, i.e. satisfy the
relationship . Prove that at least one of the three integers must
be even.
1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If...
1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If by a sequence of row operations applied to A we
reach a matrix whose last row is 0 (all entries are 0) then:
a. a,b,c,d are linearly dependent
b. one of a,b,c,d must be 0.
c. {a,b,c,d} is linearly independent.
d. {a,b,c,d} is a basis.
2. Suppose a, b, c, d are vectors in R4 . Then they form a...