Please prove the following formally and clearly:

Let X₁ = 1. Define X_n+1 = sqrt(3 + X_n). Show that (X_n) is convergent and find its limit.

A coffeehouse sells a pound of coffee for ​$8.25. Expenses are ​$3 comma 500 each​ month, plus ​$3.50 for each pound of coffee sold. ​(a) Write a function​ R(x) for the total monthly revenue as a function of the number of pounds of coffee sold. ​(b) Write a function​ E(x) for the total monthly expenses as a function of the number of pounds of coffee sold. ​(c) Write a function ​(Rminus​E)(x) for the total monthly profit as a function of the number of pounds of coffee sold. ​(a) The revenue function is ​R(x)equals nothing. ​(Simplify your answer. Use integers or decimals for any numbers in the expression. Do not include the​ $ symbol in your​ answer.) ​(b) The expense function is ​E(x)equals nothing. ​(Simplify your answer. Use integers or decimals for any numbers in the expression. Do not include the​ $ symbol in your​ answer.) ​(c) The profit function is ​(Rminus​E)(x)equals nothing. ​(Simplify your answer. Use integers or decimals for any numbers in the expression. Do not include the​ $ symbol in your​ answer.)

Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of partitions of n + 3 into exactly two distinct parts

x^2 y ′′ + xy′ + λy = 0 with y(1) = y(2) and y ′ (1) = y ′ (2)

Please find the eigenvalues and eigenfunctions and eigenfunction expansion of f(x) = 6.

4. Let a < b and f be monotone on [a, b]. Prove that f is Riemann integrable on [a, b].

A cohort study was undertaken to examine the association between high lipid level and coronary heart disease (CHD). Participants were classified as having either a high lipid level (exposed) or a low or normal lipid level (unexposed). Because age is associated with both lipid level and risk of heart disease, age was considered a potential confounder or effect modifier and the age of each subject was recorded. The following data describes the study participants: Overall, there were 11,000 young participants and 9,000 old participants. Of the 4,000 young participants with high lipid levels, 20 of them developed CHD. Of the 6,000 old participants with high lipid levels, 200 of them developed CHD. In the unexposed, 18 young and 65 old participants developed CHD.. Calculate the appropriate crude ratio measure of association combining the data for young and old individuals. 3. Now, perform a stratified analysis and calculate the appropriate stratum-specific ratio measures of association. What are they? 4. Do the data provide evidence of effect measure modification on the ratio scale? Justify your answer.

discrete mathematics

1. 

Show that if a | b and b | a, where a and b are integers, then a = b or a = -b.  

//Ex. 5, Page 208.

2.

Show that if a, b, c, and d are integers such that a | c and b | d, then ab | cd.

//Ex. 6, Page 208. 

3.

What are the quotient and remainder when

a) 44 is divided by 8?

b) 777 is divided by 21?

f) 0 is divided by 17?

g) 1,234,567 is divided by 1001?   

//(a), (b), (f), and (g) Ex. 10, Page 209.

4.

Determine whether each of these integers is a prime.

a) 19

b) 27

e) 107

f) 113

//(a), (b), (e), and (f) Ex. 2, Page 217.

5.

Find the prime factorization of each of these integers.

a) 39

c) 101

d) 143

f) 899

//(a), (c), (d), and (f) Ex. 4, Page 217.

6.

What are the greatest common divisors of these pairs of integers?

a) 23 * 33 * 55 and 25 * 33 * 52

d) 22 * 7 and 53 * 13 

//(a) and (d) Ex. 20, Page 218.

7.

Find gcd(1000, 625) and lcm(1000, 625) and verify that gcd(1000, 625)*lcm(1000, 625) = 1000*625 

//Ex. 24, Page 218.

Show that a graph T is a tree if and only if for every two vertices x, y ∈ V (T), there exists exactly one path from x to y.

Draw a sketch of z ∈ C | Im((3 − 2i)z) > 6 .

2. Express z = −i − √ 3 in the form r cis θ where θ = Argz.

3. Use de Moivre’s theorem to find the two square roots of −4i.

A stock expects to pay a dividend of $3.72 per share next year. The dividend is expected to grow at 25 percent per year for three years followed by a constant dividend growth rate of 4 percent per year in perpetuity. What is the expected stock price per share 5 years from today, if the required return is 12 percent?

Solve the following differential equation using the power series method.

(1+x^2)y''-y'+y=0

Residents were surveyed in order to determine which flowers to plant in the new public garden. A total of N people participated in the survey. Exactly 9/14 of those surveyed said that the color of the flower was important. Exactly 7/12 of those surveyed said that the smell of the flower was important. In total, 753 people said that both the color and smell were important. How many possible values are there for N?

Please explain all the steps clearly.
Thank you.

Definition 2.3.2 in our book defines the power set of a set S, denoted by P(S), as the set of all subsets of S, that is P(S) = {A : A ⊆ S}. For example, P({1, 2}) = {∅, {1}, {2}, {1, 2}}, and P(∅) = {∅}. Consider the relation ⊆ on the power set P(Z), i.e. the is-a-subset-of relation defined on sets of integers. In other words, the objects we compare are sets of integers. (a) Prove or disprove: ⊆ is reflexive. (b) Prove or disprove: ⊆ is irreflexive. (c) Prove or disprove: ⊆ is symmetric. (d) Prove or disprove: ⊆ is antisymmetric. (e) Prove or disprove: ⊆ is transitive. (f) Is ⊆ on Z an equivalence relation? Is it a partial order? (g) Which other relation satisfies exactly the same properties?