Question

Consider a sequence defined recursively as X_{0}=
1,X_{1}= 3, and X_{n}=X_{n-1}+
3X_{n-2} for n ≥ 2. Prove that X_{n}=O(2.4^n) and
X_{n} = Ω(2.3^n).

Hint:First, prove by induction that 1/2*(2.3^n) ≤ X_{n}
≤ 2.8^n for all n ≥ 0

Find claim, base case and inductive step. Please show step and explain all work and details

Answer #1

Any doubt in any step then comment below...i will explain you okk..

In solution i use Tn in place of Xn...so dont be confuse , both are same ..ok....

I prove that relation one by one , okk...

Base step is for n= 9 ..

And then we assume that result is true for any general step n= k ..and then by using this we have to prove that result is true for n= k+1 .. this is inductive step...

Define a sequence (xn)n≥1 recursively by x1 = 1, x2 = 2, and xn
= ((xn−1)+(xn−2))/ 2 for n > 2. Prove that limn→∞ xn = x exists
and find its value.

Define a sequence (xn)n≥1 recursively by x1 = 1 and
xn = 1 + 1 /(xn−1) for n > 0. Prove that limn→∞ xn = x exists
and find its value.

Suppose that a sequence an (n = 0,1,2,...) is defined
recursively by a0 = 1, a1 = 7, an = 4an−1 − 4an−2 (n ≥ 2). Prove by
induction that an = (5n + 2)2n−1 for all n ≥ 0.

Assume S is a recursivey defined set, defined by the following
properties:
1 ∈ S
n ∈ S ---> 2n ∈ S
n ∈ S ---> 7n ∈ S
Use structural induction to prove that all members of S are
numbers of the form 2^a7^b, with a and b being non-negative
integers. Your proof must be concise.
Remember to avoid the following common mistakes on structural
induction proofs:
-trying to force structural Induction into linear Induction.
the inductive step is...

Define a Q-sequence recursively as
follows.
B.
x, 4 − x is a Q-sequence for any real number x.
R.
If x1, x2, ,
xj and y1, y2, ,
yk are Q-sequences, so is
x1 − 1,
x2, , xj, y1,
y2, , yk − 3.
Use structural induction (i.e., induction on the recursive
definition) to prove that the sum of the numbers in any
Q-sequence is 4.
Base Case: Any Q-sequence formed by the base
case of the definition has sum
x +...

. Consider the sequence defined recursively as a0 = 5, a1 = 16
and ak = 7ak−1 − 10ak−2 for all integers k ≥ 2. Prove that an = 3 ·
2 n + 2 · 5 n for each integer n ≥ 0

Suppose that the sequence x0, x1, x2... is defined by x0 = 3, x1
= 7, and xk+2 = xk+1+20xk for k?0. Find a general formula for
xk.
I don't even know how to start this.
Thanks!

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 .

Suppose that the sequence x0,
x1, x2... is defined by
x0 = 6, x1 = 5, and
xk+2 =
?3xk+1?2xk for
k?0. Find a general formula for xk. Be
sure to include parentheses where necessary, e.g. to distinguish
1/(2k) from 1/2k. .
xk
= ?

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

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