Question

. 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

Answer #1

We are going to prove by strong induction, that the given recursive sequence satisfies .

For n=0, and for n=1, .

Now suppose, the given recursive sequence satisfies the answer for all , that is, for all for some . Then, .

Hence, we have proved by induction that the given recursive sequence satisfies .

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.

2. Exercise 19 section 5.4. Suppose that a1, a2, a3, …. Is a
sequence defined as follows:
a1=1 ak=2a⌊k/2⌋ for every integer k>=2.
Prove that an <= n for each integer n >=1.
plzz send with all the step

Consider the sequence defined recursively by
an+1 = (an + 1)/2 if an is an odd number
an+1 = an/2 if an is an even number
(a) Let a0 be equal to the last digit in your student number,
and compute a1, a2, a3, a4.
(b) Suppose an = 1, and find an+4.
(c) If a0 = 4, does limn→∞ an exist?

Consider a sequence defined recursively as X0=
1,X1= 3, and Xn=Xn-1+
3Xn-2 for n ≥ 2. Prove that Xn=O(2.4^n) and
Xn = Ω(2.3^n).
Hint:First, prove by induction that 1/2*(2.3^n) ≤ Xn
≤ 2.8^n for all n ≥ 0
Find claim, base case and inductive step. Please show step and
explain all work and details

1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . .
, ak) = 1, i.e., the largest
positive integer dividing all of a1, . . . , ak is 1. Prove that
the equation
a1u1 + a2u2 + · · · + akuk = 1
has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly
apply the extended Euclidean
algorithm, Theorem 1.11. You may find it easier to prove...

4.
Let an be the sequence defined by a0 = 0 and an = 2an−1 + 2 for
n > 1.
(a) Find the value of sum 4 i=0 ai .
(b) Use induction to prove that an = 2n+1 − 2 for all n ∈ N.

10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be integers and p be a prime. If p|(a1 · a2 · a3 · · · ak), then
p|ai for some i with 1 ≤ i ≤ k.” Prove that P(k) holds for all
positive integers k

1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and an = 4an−1. Guess a formula for an and prove that your
guess is correct.
2) Show that given 5 integer numbers, you can always find two of
the numbers whose difference will be a multiple of 4.
3) Four cats and five mice form a row. In how many ways can they
form the row if the mice are always together?
Please help...

Let (a_n)∞n=1 be a sequence defined recursively by a1 = 1, a_n+1
= sqrt(3a_n) for n ≥ 1. we know that the sequence converges. Find
its limit.
Hint: You may make use of the property that lim n→∞ b_n = lim
n→∞ b_n if a sequence (b_n)∞n=1 converges to a positive real
number.

Given this pseudocode:
input: sequence of numbers ak
k, length of sequence
answer := a1
for i = 2 to k
if (ai > answer), then answer = ai
End-for
What is the value of answer for the sequence {-1, 4, -7, 10, 2}
with k = 5?
Please provide detailed answer! Thank you!

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