Question

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 me with these and provide all the work done to reach the answer.

Answer #1

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

Let A = (A1, A2, A3,.....Ai) be defined as a sequence containing
positive and negative integer numbers.
A substring is defined as (An, An+1,.....Am) where 1 <= n
< m <= i.
Now, the weight of the substring is the sum of all its
elements.
Showing your algorithms and proper working:
1) Does there exist a substring with no weight or zero
weight?
2) Please list the substring which contains the maximum weight
found in the sequence.

1. (24 pts.) For which integer values of the constant k does the
sequence {a1, a2, a3, . . .} defined by
an = (5n^k + 3)/(3n^k + 5)
converge? For those values of k, compute the limit of the
sequence {a1, a2, a3, . . .}.

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...

Events A1, A2, and A3 form a partiton of sample space S with
Pr(A1)=27, Pr(A2)=47, and Pr(A3)=17. E is an event in S
with Pr(E|A1)=35, Pr(E|A2)=25, and Pr(E|A3)=15.
What is Pr(E)?
What is Pr(A1|E)?
What is Pr(E′)?
What is Pr(A′1|E′)?
Enter your answers as whole numbers or fractions in lowest
terms.

Events A1,A2, and A3 form a partition of sample space
S with Pr(A1)=3/7, Pr(A2)=3/7, Pr(A3)=1/7. E is an event in S with
Pr(E|A1)=3/5, Pr(E|A2)=2/5, and Pr(E|A3)=3/5.
What is Pr(E)?
What is Pr(A2|E)?
What is Pr(E')?
What is Pr(A2'|E')?

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.

. For any integer n ≥ 2, let A(n) denote the number of ways to
fully parenthesize a sum of n terms such as a1 + · · · + an.
Examples:
• A(2) = 1, since the only way to fully parenthesize a1 + a2 is
(a1 + a2).
• A(3) = 2, since the only ways to fully parenthesize a1 + a2 +
a3 are ((a1 + a2) + a3) and
(a1 + (a2 + a3)).
• A(4)...

. 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

Find a general term (as a function of the variable n) for the
sequence{?1,?2,?3,?4,…}={45,1625,64125,256625,…}{a1,a2,a3,a4,…}={45,1625,64125,256625,…}.
Find a general term (as a function of the variable n) for the
sequence {?1,?2,?3,?4,…}={4/5,16/25,64/125,256/625,…}
an=
Determine whether the sequence is divergent or convergent. If
it is convergent, evaluate its limit.
(If it diverges to infinity, state your answer as inf . If it
diverges to negative infinity, state your answer as -inf . If it
diverges without being infinity or negative infinity, state your
answer...

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