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Common ratio of geometric series

Use the geometric sequence of numbers 1, 3, 9, 27, ... to find the following: a) What is r, the ratio between 2 consecutive terms? b) Using the formula for the nth term of a geometric sequence, what is the 10th term? c) Using the formula for the sum of a geometric series, what is the sum of the first 10 terms?

Sequences and Series

Using the index of a series as the domain and the value of the series as the range, is a series a function? Include the following in the answer: Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic series? Which one of the basic functions (linear, quadratic, ration

Cauchy Sequences

Consider the real number iteration scheme x_n+1 = f(x_n) for n = 1, 2, ... with x_1 given. In addition, suppose there is a number 0 < p < 1 st lf(x) - f(y)l < = plx-yl for all x,y. a) Show lx_n+1 - x_nl < = p^n-1lx_2 - x_1l for all n. b) From this, conclude {x_n}_n is Cauchy.

Proof: Sequences and Supremum

Suppose that the sequences {a_n}_n is bounded above and lim(b_n) exists. a) Prove that for all e>0 there is an N st that for all n>=N sup{a_k:k>=n} + b_n <=sup{a_k + b_k: k>=n} + e. b) Use this to conclude limsup(a_n) + lim (b_n) <= limsup (a_n+b_n) (all limits are n ---> infinit

Proofs : Bounded Sequences

Suppose the sequences {a_n}_n and {b_n}_n are both bounded above. a) Prove that for all n in the naturals sup{a_k + b_k: k>/=n} is less than or equals sup{a_k:k>/=n} + sup{b_k:k>/=n} b) Use this to conclude: limsup (a_n + b_n) is less than or equals limsup(a_n) + limsup(b_n) (all limits are n--> infinity)

For the equation x - 2(sqrt of x) = 0 perform the following:

For the equation x - 2(sqrt of x) = 0 perform the following: Solve for all values of x that satifies the equation Graph the functions y = x and y = 2(sqrt of x) on the same graph. Show the intersection of these two graphs. Please show your work to help me with the more difficult equations on my assignment. Thanks

Initial and Future Value

You have decided to purchase some shares of stock for $1000. After five years, the value of your purchase has grown to $2000. a. Write a formula for the relationship between the future value of an investment and the initial investment amount. Use variables instead of actual quantities in your formula. Note what each variable

Average Annual Return on Investment Formula

The average annual return for an investment is given by the formula r= (s/p)^1/n -1 where p is the initial investment and s is the amount it is worth after n years. The top mutual fund for 1997 in the 3-year category was Fidelity Select-Energy Services, in which and investment of $10,000 grew to $31,895.06 from 1994 to 1997

Problem Set

11. Assume the total number of dollars (in billions) on entertainment in the United States from 1990 to 2000 can be approximated by the model S = 148 + 6.7t + 0.58t2. where t = 0 represents the year 1990. During which year was $215 billion the amount spent on entertainment? (5 points) 12. Solve 3x2 - 10x = 8 by factoring

Find the constant term in the expansion (1/(2x^3) + x)^20.

Find the constant term in the expansion of (1/(2x^3) + x)^20. [The constant term is the summand which does not involve any power of x. For example the constant term in 3x^2&#8722; 4x + 23 +9x^10 + 5/(3x^6) is 23.] [Note on notation: 2^3 means '2 to the power of 3', so 2^3 = 8] [The pdf file contains the question in pr

Transaction between Achmet and Ali

13. Achmed and Ali were camel - drivers but one day they decided to quit their job. they wanted to become shepherds. They went to the market and sold all their camels. The amount of money (dinars) they received for each camel was the same as the total amount of camels they owned. With that money they bought as many sheep as

select the best model.

(See attached file for full problem description) --- 1. The following gives the price in dollars of a round trip ticket from Phoenix to Portland, and the corresponding profit, in millions of dollars, for each price. For the data below, select the best model. Price (dollars) 200 250 300 350 400 450 3.09 3.51

Theory of Computation : Non-deterministic finite state automaton

Give a non-deterministic finite state automaton that recognizes the language L subset {0,1}* that consists of all words w such that some appearance of two 0's in w is separated by an even number of 1's. (For example: the words 0010010110, 001101 are in the language but the words 0101110, 001001 are not.) (As a note, this i

Writing Equations from Word Problems : Time and Distance, Two Moving Objects

At 2:00 p.m. bike A is 4km north of point C and traveling south as 16km/h. At the same time, bike B is 2 km east of C and traveling east at 12km/h. a. Show that t hours after 2:00 p.m. the distance between the bikes is: square root (400t^2 - 80t + 20) b. At what time is the distance between the bikes the le

2 Problems

(See attached file for full problem description) --- 1. Make y the subject of the formula (see attached) 2. Rearrange the equation X = 1/2A In(q - 3) + c to obtain a formula for q

Series and Annuities: The Tortoise and the Hare Series

You may have heard the fable about the tortoise and the hare. Suppose that the tortoise and the hare are running a 5000 M race. The tortoise proceeds very slowly, never changing its speed. The hare runs very quickly at the start. The tortoise travels 10 M every minute. the hare travels 2500 M in the first minute but, in each mi

Word problems

Please help me understand how to solve these 2 word problems: 1.) McMoRan projects that in 2010 world grain supply will be 1.8 trillion metric tons and the supply will be only 3/4 of world grain demand. What will world grain demand be in 2010? 2.) Before Ronald sold two female rabbits, half of his rabbits were femal

SQ Root

For the equation x-the sq root of x=0, how do I solve for all values of x to satisfy the equation?

Square roots

Can you show me how to solve the following equations? Square roots confuse me. Sq Root of X-1=3? Sq Root of X^3=8? and ^3sq root of x^2=4? Is the sq root of x^2=x an identity?

Linear Function of Distance and Time

How do I write a linear function showing distance traveled, d, as a function of time, t, ? I know the miles are 300 and the car is moving at a constant speed of 60mph. I know if you divide 300 by 60 the car will arrive at it's destination in 5 hours and at 3 hours it will have only traveled 180 miles. But how do I show this ?

Simple and Compound Interest

You have $380,000 in your bank account. The interest rate in your account is 5%. Solve for the following: a. How much interest will you accumulate if interest is compounded annually over the next five years? b. How much interest will you earn in your account over the next five years with continuous compounding?

Fibonacci sequence

The problem is Take any of the generalizations about the Fibonacci Numbers that we have considered, and investigate what happens if the sequence is formed by two different starting numbers but continues in the same way by adding successive pairs of terms. For example, you might form a new pseudo-Fibonacci sequence in this w

L'Hospital's Rule, Asymptotes, Global Extrema, Inflection Points

F(x) = x^2 e^(17x) 1. Find an equation for each horizontal asymptote to the graph of f. 2. Find an equation for each vertical asymptote to the graph of f. 3. Determine all critical numbers. 4. Determine the global maximum of the function. 5. Determine the global minimum of the function. 6. Find inflection points. 7. Fin

Real and Non-Real Affine Intersection Points and Their Multiplicities

For the first 5 questions, consider the set of intersection points of two equations, and let a1 be the number of distinct affine real intersections with multiplicity one, let a2 be the number of distinct affine real intersections with multiplicity two, let b be the numberof distint complex non-real affine intersections and let

Splitting Fields : Find the splitting fields over Q for x^3+3x^2+3x-4.

Find the splitting fields over Q for x^3+3x^2+3x-4. Recall a splitting field is as follows: Let K be a field and let f(x)=a_0+a_1*x+...+a_n*x^n be a polynomial in K[x] of degree n>0. An extension field F of K is called a splitting field for f(x) over K if there exist elements r_1,r_2,...,r_n elements of F such that (i) f(x