1a. Consider the sequence {?? }n≥0 which starts 1,2,7,20,61,122,..., defined by the recurrence relation ?? = 2??−1 + 3??−2 and initial conditions ?0 = 1, ?1 = 2. Solve the recurrence relation. That is, find a closed formula for ??. Show your work.

The abandoned field behind your house is home to a large prairie dog colony. Each week the size of the colony triples. However, sadly 4 prairie dogs die each week as well (after the tripling occurs). Consider the sequence ?0, ?1, ,a2,..., where ?? is the number of prairie dogs in the colony after n weeks.

(b) Write down a recurrence relation to describe an and briefly explain.

(c) Explain why if ?? is even, then ??+1 must also be even.

(d) Suppose you wanted to prove by mathematical induction that an was always even. What would the base case be and why is it needed? Your answer should be specific to this context.

(e) Your friend believes what you have written in parts (b) and (c), but still does not see why ?3 must be even because he does not understand the logic behind induction. Explain why induction in this case proves that there will be an even number of prairie dogs in week 3 specifically

Please with clear legible hand writing, and number your work.

What solutions did nineteenth-century analysts like Cauchy and Weierstrass find to the philosophical difficulties connected with infinitesimals?

certain examples if derivatives in Economics and Commerce

Let G be a group and a be an element of G. Let φ:Z→G be a map defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is a subgroup ofG.

Solve the system of linear equations using the Gauss-Jordan elimination method.

2x + 3y - 2z = 8

3x - 2y + 2z = 2

4x - y + 3z = 2

(x, y, z) = ?

Give an example for each of the following. Justify your answers.

i) an infinite subset of E¹ with no cluster points. (consider Z)

ii) a complete metric space that is bounded but not compact (hint: consider the trivial metric space (S,d) with S being an infinite set).

In each of the following, show that ? is a subgroup of ?. (1)

1. ? = 〈F(R), +〉, ? = {? ∈ F(R): ?(?)=0 for every ?∈[0,1]}.

2. ? = 〈F(R), +〉, ? = {? ∈ F(R): ?(?)=−?(?)}.

Do the following problems:

a. Use mathematical induction to show that 1 + 2 + 22 +···+ 2n = 2n+1 − 1 for all non-negative integers n.
b. A coin is weighted so that P(H) = 2/3 and P(T) = 1/3. The coin is tossed 4 times. Let the random variable X denote the number of heads that appear. (x) Find the distribution of X; (xx) Find the expectation E(X).
c. Show a derivation of Bayes’ Theorem

Select one improvement program that you have observed in an organization. The program of improvement may focus on quality, safety, absenteeism, costs, or other matters. With respect to implementation the program:
A) What where its strength?
B) What were the weaknesses in implementation?
C) What recommendations would you make to implement an improvement program in the future?

                                                           Question Two

Faulu College has a total of 52 students undertaking three courses namely; Business Mathematics, Economics and Law. During recently conducted examinations, 13 students excelled in Business Mathematics and Law, 16 students excelled in Business Mathematics and Economics and 12 students excelled in Law and Economics. The number of students that excelled in Economics was 24 while 2 students excelled in none of the three courses. The number of students who excelled in Business Mathematics only was twice the number of students who excelled in Law only. The number of students who excelled in Law only was six times the number of students who excelled in Economics only.

Required:

1. Present the above information in the form of a venn diagram          
2. The number of students who excelled in all the three courses          
3. The number of students who excelled in two courses only               
4. The number of students who excelled in one course only                 

Question three

Parents of a young girl want to deposit a sum of money which will earn interest at the rate of 9% per year compounded semiannually. The deposit will be used to generate a series of eight semiannual payment of Sh. 2500 beginning 6 months after the deposit. These payments will be used to help finance their daughter’s high school education.

Required:

1. What amount must be deposited to achieve their goal?                  
2. How much interest will be earned on this deposit?                         (3marks)

Can you please give an original example of a application integration hierarchy and explain it for me; you can give one or up to 3 design examples (i.e., data sharing, functionality sharing, or process coordination).

The One-Way ANOVA applet lets you see how the F statistic and the P-value depend on the variability of the data within groups, the sample size, and the differences among the means.

(a)

The black dots are at the means of the three groups. Move these up and down until you get a configuration that gives a P-value of about 0.01. What is the value of the F statistic?

(b)

Now increase the variation within the groups by sliding the standard deviation bar to the right. Describe what happens to the F statistic and the P-value.

c)

Using between- and within-group variation, explain why the F statistic and P-value change in this way.

The One-way applet:

Std.Dev. =1.95, Mean =6.08

Std. Dev. =1.85, Mean=4.21

Std.Dev. =1.84, Mean=4.86

We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 different “digits” {0,1,...,9}. Sometimes though it is useful to write numbers in hexadecimal or base 16. Now there are 16 distinct digits that can be used to form numbers: {0,1,...,9,A,B,C,D,E,F}. So for example, a 3 digit hexadecimal number might be 3B8. (a) How many 2-digit hexadecimals are there in which the first digit is E or F? Explain your answer in terms of the additive principle (using either events or sets). (b) Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer? (c) How many 3-digit hexadecimals start with a letter (A-F) and end with a numeral (0-9)? Explain. ( d) How many 3-digit hexadecimals start with a letter (A-F) or end with a numeral (0-9) (or both)? Explain.

Suppose the lengths of human pregnancies are normally distributed with muμequals=266266 days and sigmaσequals=1616 days. Complete parts ​(a) and​ (b) below. ​(a) The figure to the right represents the normal curve with mu equals 266μ=266 days and sigmaσequals=1616 days. The area to the leftleft of Upper X equals 240X=240 is 0.05210.0521. Provide two interpretations of this area. Provide one interpretation of the area using the given values. Select the correct choice below and fill in the answer boxes to complete your choice. ​(Type integers or​ decimals.) A. The proportion of human pregnancies that last moremore than nothing days is nothing. B. The proportion of human pregnancies that last lessless than nothing days is nothing. X font size decreased by 3 266266 font size decreased by 3 240240 A normal curve has a horizontal axis labeled "X" and two horizontal coordinates, 240 and 266. The curve's peak is near the top of the graph at horizontal coordinate 266. Two vertical line segments run from the horizontal axis to the curve at 240 and 266. The area under the curve to the left of 240 is shaded. Provide a second interpretation of the area using the given values. Select the correct choice below and fill in the answer boxes to complete your choice. ​(Type integers or​ decimals.) A. The probability that a randomly selected human pregnancy lasts lessless than nothing days is nothing. B. The probability that a randomly selected human pregnancy lasts moremore than nothing days is nothing. ​(b) The figure to the right represents the normal curve with mu equals 266μ=266 days and sigmaσequals=1616 days. The area between xequals=280280 and x equals 295x=295 is 0.15580.1558. Provide two interpretations of this area. Provide one interpretation of the area using the given values. Select the correct choice below and fill in the answer boxes to complete your choice. ​(Type integers or decimals. Use ascending​ order.) A. The proportion of human pregnancies that last between nothing and nothing days is nothing. B. The proportion of human pregnancies that last less than nothing or more than nothing days is nothing. X font size decreased by 3 266266 font size decreased by 3 280280 font size decreased by 3 295295 A normal curve has a horizontal axis labeled "X" and three horizontal coordinates, 266, 280, and 295. The curve's peak is near the top of the graph at horizontal coordinate 266. Three vertical line segments run from the horizontal axis to the curve at 266, 280, and 295. The area under the curve between the vertical line segments at 280 and 295 is shaded. Provide a second interpretation of the area using the given values. Select the correct choice below and fill in the answer boxes to complete your choice. ​(Type integers or decimals. Use ascending​ order.) A. The probability that a randomly selected human pregnancy lasts between nothing and nothing days is nothing. B. The probability that a randomly selected human pregnancy lasts less than nothing or more than nothing days is nothing.

a. Find the x-value of each point under the given translations : (-1,-6) with a horizontal translation of 5 units

b. Find the y value of the image of the point under the translation: (5,-5) with a vertical translation of 3 units

c. The graph of a particular function contains the point (-2,6). An image of that function has a coordinate at (3,-5). What is the rule for this translation?

d. Find the image of (5,-3) under the same translation?

e. The graph of f(x) = |x| is translated 3 units down and 6 units to the left. Write an equation for the image.

f. The equation x² + y² = 25 describes a circle of radius 5 with center at the origin.  Describe the graph of the relation (x + 2)² + (y + 9)² = 25 . Where is the new center of the circle and what is the radius of that circle?