Consider the sequences 2,5, 12, 29, 70, 169, 408, ... (with ao = 2) a. Describe the rate of growth of this sequence. b. Find a recursive definition for the sequence. c. Find a closed formula for the sequence. d. If you look at the sequence of differences between terms, and then the sequence of second differences, the sequence of third differences, and so on, will you ever get a constant sequence? Explain how you know.
Answers
a) The rate of growth of the sequence is \(r=(1\pm \sqrt2)\).
b) A recursive definition for the sequence is \(a_n=c_1(1+\sqrt2)^n+c_2(1-\sqrt2)^n\).
c) A closed formula for the sequence is \(a_n=\frac{(\sqrt2+1)^{n+2}-(1-\sqrt2)^{n+2}}{2\sqrt2}}\).
Given that, 2, 5, 12, 29, 70, 169, 408.
\(2, 2*2+1, 5\times2+2, 12\times2+5, 29\times2+12, 70\times2+29, 169\times2+70,.......(2\times a_{n-1}+a_{n-2})\)
Clearly, if we take first four terms as \(a_0, a_1, a_2, a_3\)
\(2, 2\times2+2, 5\times2+2, 12\times2+5\)
\(a_3=2a_1+a_0\)
\(a_4=2a_3+a_2\)
\(a_2=2a_2+a_0\)
\(a_n=2a_{n-1}+a_{n-2}\) with \(a_0=2, a_1=5\)
b) Growth rate: Sequence has exponential growth rate.
Solving the sequence for closed form,
\(a_n-2a_{n-1}-a_{n-2}=0\)
\((r^2-2r-1)=0\)
\((r-1)^2+(-2)=0\)
\((r-1)^2=2\)
\(r-1=\pm\sqrt2\)
\(r=(1\pm \sqrt2)\)
c) General solution:
\(a_n=c_1(r_1)^n+c_2(r_2)^n\)
\(a_n=c_1(1+\sqrt2)^n+c_2(1-\sqrt2)^n\)
Using \(a_0=2 \imples c_1+c_2=2 -----(i)\)
\(a_1=5 \implies 5=(\sqrt2+1)c_1+(1-\sqrt2)c_2 ------(ii)\)
Solving (i) and (ii), we get
\(c_1=\frac{3+2\sqrt2}{2\sqrt2} \ and \ c_2=\frac{2\sqrt2-3}{2\sqrt2}\)
\(a_n=c_1(1+\sqrt2)^n+c_2(1-\sqrt2)^n\)
\(= \frac{(3+2\sqrt2)}{2\sqrt2}(1+\sqrt2)^n+\frac{(2\sqrt2-3)}{2\sqrt2}(1-\sqrt2)^n\)
\(= \frac{(\sqrt2+1)^2}{2\sqrt2}(1+\sqrt2)^n-\frac{(\sqrt2-1)^2}{2\sqrt2}(1-\sqrt2)^n\)
\(a_n=\frac{(\sqrt2+1)^{n+2}-(1-\sqrt2)^{n+2}}{2\sqrt2}}\)
Therefore,
a) The rate of growth of the sequence is \(r=(1\pm \sqrt2)\).
b) A recursive definition for the sequence is \(a_n=c_1(1+\sqrt2)^n+c_2(1-\sqrt2)^n\).
c) A closed formula for the sequence is \(a_n=\frac{(\sqrt2+1)^{n+2}-(1-\sqrt2)^{n+2}}{2\sqrt2}}\).
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Related Questions
Dax is buying coffee for 5 5 people in his office. He also leaves a $2.00 $ 2.00 tip for the barista. If his total, with tip, is $18.25, $ 18.25 , how much is each cup of coffee, not including the tip? Enter your answer as a decimal, like this: 42.53
Answers
Answer:
$3.25
Step-by-step explanation:
To get the price of one cup of coffee without including Dax's two dollar tip, the first step is to subtract 2 from 18.25.
18.25 - 2= 16.25
16.25 is price of the five cups of coffee that Dax bought without the two dollar tip. The final step is to get the price of one cup of coffee which is basically:
16.25 ÷ 5 = 3.25
The cost of each cup of coffee is $3.25.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
To get the price of one cup of coffee without including Dax's two-dollar tip, the first step is to subtract 2 from 18.25.
18.25 - 2= 16.25
16.25 is the price of the five cups of coffee that Dax bought without the two-dollar tip. The final step is to get the price of one cup of coffee which is basically:
16.25 ÷ 5 = $3.25
Hence, the cost will be $3.25.
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The inequality x ≤ -8 would represent which of the following values?
{8, 5, 0}
{-6, -7, -8}
{-10, -9, -8}
None of these choices are correct.
Answers
Answer:
{-10. -9, -8}
Step-by-step explanation:
x ≤ -8 would be all values of x less than or equal to -8
x = -8, -9, -10
X has the following probability distribution.
X -2 -1 9 1 2
P(X) .2 .2 .2 .2 .2
Compute the expected value of X.
Select one:
a. 1.0
b. 2.4
c. 1.3
d. 1.8
Answers
In this case, using the given probability distribution for X, we calculate the expected value as 1.8.
To compute the expected value of a random variable X, we multiply each possible value of X by its corresponding probability and sum up the results.
Given the probability distribution for X:
X: -2 -1 9 1 2
P(X): 0.2 0.2 0.2 0.2 0.2
To calculate the expected value, we perform the following calculations:
(-2 * 0.2) + (-1 * 0.2) + (9 * 0.2) + (1 * 0.2) + (2 * 0.2)
= (-0.4) + (-0.2) + (1.8) + (0.2) + (0.4)
= 1.8
Therefore, the expected value of X is 1.8.
The correct answer is d. 1.8.
To compute the expected value of a random variable, we multiply each value of the random variable by its corresponding probability and sum up the results.
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(HELP me PlZ If u do ill five u a pringle >:)
If the corner that the lawyer’s office is on measures 78°, what is the measure of the corner where the clothing store is located? What word describes this relationship?
Answers
They are congruent, therefore both are 78°
Help?
Canyon middle school has 1,250 students. 40% of these students ride the bus. How many students ride the bus?
Answers
Answer:
500 students
Step-by-step explanation:
Change the percent to a fraction in its simplest terms
Then solve 2/3×1250
Answer:
500 students ride the bus
Step-by-step explanation:
Easy:
If the total number of students is 1250 and you know 40% ride the bus all you have to do is figure out 40% of 1250. Once you get that you will get 500 people. You can use a calculator for this problem.
x/1250 = 40/100
x = number of students riding bus
Solve for x
100x = 1250*4 - after you cross mulitply
solve for x
x = 500
Rewrite in simplest terms: (4x+5)+(−5x−1) pls help
Answers
Answer:
-x + 4
Step-by-step explanation:
(4x+5)+(−5x−1)
4x+5−5x−1 [Remove the parentheses]
-x + 4 [Combine Like Terms]
Answer:
-x+4
Step-by-step explanation:
Given:
(4x+5)+(−5x−1)
Solution:
To re-write this in it's simplest terms:
4x+5-5x−1Combine like terms:
4x-5x+5-1-x+4Hence the simplest form is -x+4.
Done.
in the diagram the measure of angle one is 3X the measure of angle two is 12 X what is the measure of angle 4
Answers
Answer:
144°
Step-by-step explanation:
m<1 =3x
m< 2= 12x
1&2 are supplementary angles, so
3x+12x= 180
15x=180
x= 12°
measure of angle 4 equals the measure of angle 2
because they're head opposite angles
m<4= 144°
Rashaad went to the store to buy some apples. The price per pound of the apples is $4.75 per pound and he has a coupon for $2.50 off the final amount. With the coupon, how much would Rashaad have to pay to buy 5 pounds of apples? Also, write an expression for the cost to buy
p
p pounds of apples, assuming at least one pound is purchased.
Answers
Answer:
21.75
Step-by-step explanation:
sorry if im wrong
Pls someone help. I have a test tomorrow
Answers
Answer:
Step-by-step explanation:
solve the given initial-value problem. y'' 49y = 0, y(0) = 4, y'(0) = −5
Answers
Particular solution with the determined constants is:
y(t) = 4*cos(7t) - (5/7)*sin(7t)
To solve the given initial-value problem y'' + 49y = 0 with initial conditions y(0) = 4 and y'(0) = -5, follow these steps:
Step 1: Identify the type of problem.
This is a second-order linear homogeneous differential equation with constant coefficients.
Step 2: Write down the characteristic equation.
The characteristic equation for this problem is \(r^2\) + 49 = 0.
Step 3: Solve the characteristic equation.
\(r^2\) = -49
r = ±√(-49) = ±7i
Since the roots are complex conjugates, the general solution of the differential equation is in the form:
y(t) = C1*cos(7t) + C2*sin(7t)
Step 4: Apply the initial conditions to find the constants.
Apply the initial condition y(0) = 4:
4 = C1*cos(0) + C2*sin(0)
4 = C1
So, C1 = 4
Next, find the first derivative of y(t):
y'(t) = -7*C1*sin(7t) + 7*C2*cos(7t)
Apply the initial condition y'(0) = -5:
-5 = -7*4*sin(0) + 7*C2*cos(0)
-5 = 7*C2
So, C2 = -5/7
Step 5: Write the particular solution.
The particular solution with the determined constants is:
y(t) = 4*cos(7t) - (5/7)*sin(7t)
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Examine the following table, which represents some points on a line.
x 3 0 −3 −6 −9
y −3 −2 −1 0 1
What is the equation for the line in slope-intercept form?
y=−3x−2y is equal to negative 3 x minus 2, , ,
y=3x−2y is equal to 3 x minus 2, , ,
y=−13x−2y is equal to negative 1 third x minus 2, , ,
y=−13x−6y is equal to negative 1 third x minus 6, , ,
y=−3x−6
Answers
Answer:
(c) y = −1/3x −2
Step-by-step explanation:
We notice two things in the table.
y goes up by 1 when x goes down by 3.y = -2 when x = 0 (this is the y-intercept__
The slope of the line is ...
m = (change in y)/(change in x) = +1/-3 = -1/3
Then the slope-intercept equation is ...
y = mx +b . . . . . . slope m, y-intercept b
y = -1/3x -2
Answer is C
Step-by-step explanation:
math
Find the value of b. PLEASE HELP ASAPPPP!!!!
A. 56
B. 140
C. 230
D. 65
Answers
Answer:
C
Step-by-step explanation:
b=360-2(65)=360-130=230
Answer:
230
Step-by-step explanation:
b= 360-2(65)
=360-130
=230
A breadth-first search (BFS) is a traversal that visits a starting vertex, then visits every vertex along each path starting from that vertex to the path's end before backtracking. True False
Answers
A breadth-first search (BFS) is a traversal algorithm that visits a starting vertex and then visits every vertex along each path starting from that vertex to the path's end before backtracking.
In a BFS, a queue is typically used to keep track of the vertices that need to be visited. The starting vertex is added to the queue, and then its adjacent vertices are added to the queue. The process continues until all vertices have been visited. This approach ensures that the traversal visits vertices in a breadth-first manner, exploring the vertices closest to the starting vertex first before moving on to the ones further away.
So, A breadth-first search (BFS) is a traversal algorithm that visits a starting vertex, then visits every vertex along each path starting from that vertex to the path's end before backtracking. This approach explores all vertices at the same level before moving on to the next level, ensuring a breadth-first exploration. Therefore, the statement is true.
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Need help What is 6/10 in simplest form
Answers
Answer:
3/5
Step-by-step explanation:
35.
Step-by-step explanation:
The simplest form of 610 is 35.
help.............gigfjfgjfjf
Answers
Answer: B
Step-by-step explanation :}
Consider the following rational expression: 4y + 16 y+ 4 Step 2 of 2: Find the restricted values of y, if any, for the given rational expression Answer How to enter your answer (opens in new window) 2
Answers
The given rational expression is 4y + 16 y + 4. To find the restricted values of y, we need to identify any values of y that would make the expression undefined.
In this case, the expression is in the form of a sum, so we don't have any denominators that could lead to division by zero. Therefore, there are no restricted values of y for this rational expression.
The expression 4y + 16 y + 4 is defined for all real numbers. We can evaluate it for any value of y without encountering any restrictions.
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URGENT
Which line is the graph of y = 2x - 1?
line a
line d
line c
line b
Answers
Answer:
The answer is b
Step-by-step explanation:
A shipping container will be used to transport several 120-kilogram crates across the country by rail. The greatest weight that can be loaded into the container is 27500 kilograms. Other shipments weighing 12400 kilograms have already been loaded into the container. Which inequality can be used to determine x, the greatest number of
120-kilogram crates that can be loaded onto the shipping container?
Answers
Inequality can be used to determine x, the greatest number of 120-kilogram crates that can be loaded onto the shipping container is 8000 + 120C <= 27500
What is inequality?An inequality is a relationship that allows us to contrast two or more mathematical expressions.
Knowing that;
Each carton weighs 120 kg.
The container's maximum weight when loaded is 27500 kg.
Weight of the container as it is currently loaded: 8000 kg
Now that we have merged, we have;
Solution
because 8000 grams have already been loaded kilograms
into the container
each crate weighs 120 kilograms.
so 8000 + 120C <= 27500
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Bianca took out a $2,600 unsubsidized stafford loan. she will be attending school for four years, and she wishes to have the loan paid off five years before its normal ten-year duration is finished. the loan has an interest rate of 6.2%, compounded monthly. how much will she have to pay monthly to avoid interest capitalization? a. $12.40 b. $19.29 c. $13.43 d. $17.36 please select the best answer from the choices provided a b c d
Answers
The total monthly payment would be $17.36.
Option (d) is correct.
What is the annuity payment?
An annuity payment is a fixed, regular payment made to an individual or organization over a specific period of time. It is typically used in the context of financial transactions, such as loans, investments, and insurance.
To calculate the monthly payment Bianca will have to make to avoid interest capitalization, we can use the formula for the annuity payment:
PMT = (Loan Amount * Interest Rate / (1 - (1 + Interest Rate)^(-n))) / (Interest Rate / 12)
Where:
PMT is the monthly payment
Loan Amount is the amount of the loan, in this case, $2,600
Interest Rate is the annual interest rate, in this case, 6.2%
n is the total number of payments, in this case, 5 years * 12 months/year = 60 months
Plugging in the values, we get:
PMT = (2600 * 0.062 / (1 - (1 + 0.062)^(-60))) / (0.062 / 12)
PMT = $17.36
Hence, the total monthly payment would be $17.36
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Answer: C
Step-by-step explanation:Bianca took out a $2,600 unsubsidized Stafford loan. She will be attending school for four years, and she wishes to have the loan paid off five years before its normal ten-year duration is finished. The loan has an interest rate of 6.2%, compounded monthly. How much will she have to pay monthly to avoid interest capitalization?
a.
$12.40
b.
$19.29
c.
$13.43
d.
$17.36
Please select the best answer from the choices provided
Teasia bought her mom a sweater for $18.99, her little brother a car for $5.99, and her grandpa a tie for $12.99. She had a coupon for 23% off of the entire purchase because it was the store’s 23rd year being open. How much did teasia get all of the items for?
Answers
Answer:
29.2369
Step-by-step explanation:
Add up all the stuff she bought.
Multiply by 23% or .23 then subtract what you get from that to the total of what she bought in all. There is your answer.
PLSSSS HELP!!!! WILL GIVE BRANLIEST
Answers
Jenna is surveying the number of trucks in parking lots.
One parking lot has 9 trucks and a total of 36 vehicles.
The next parking lot has 104 vehicles.
To find the number of trucks based on her survey.
The number of trucks in relation to the total vehicle is given by, = number of trucks divided by (top) total number of vehicles (bottom) =9/36 = 1/4.
The ratio of number of trucks to the total vehicle is 1/4, (based on the survey)
Given that, in the next parking lot has 104 vehicles.
Let n be the number of trucks.
we get, n/104 = 1/4n = 104/4 = 26
n =26
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06.11 Discussion-Based Assessment FULL ASSIGNMENTS
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State whether each sentence is true or false. If false, replace the underlined word or phrase to make a true sentence.
An equilateral triangle is always equiangular.
Answers
The given sentence about equilateral triangle is true.
A statement is given in the question.
We have to tell whether the statement is true or false.
What is an equilateral triangle ?
An equilateral triangle is a triangle in which all sides and all angles are equal.
As per the question ;
The statement given is , "An equilateral triangle is always equiangular."
Let's check whether it is true or false ;
We know that ,
Equiangular meant equal angles and we know that all the angles of equilateral triangle are equal.
Also ;
An equilateral triangle can also said as an isosceles triangle, that meant any two sides that are equal have equal opposite angles.
So ;
All three sides of an equilateral triangle are equal, all three angles are equal, too. We can conclude that every equilateral triangle is also equiangular.
Thus , the given statement about equilateral triangle is true.
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Solve for d.
2 = 2(d - 7)
Answers
Answer:
should be 6to make is one so you can multiply it by 1 to get the same number
Step-by-step explanation:
Answer:
D = 8
Step-by-step explanation:
distribute
2 = 2d - 14
add 14 to both side
16 = 2d
divide both sides by 2
8 = d
d = 8
can I get brainliest ?
find a polynomial function of degree 3 with real coefficients that has the given zeros -2,3,-6
Answers
A polynomial function of degree 3 with real coefficients that has the given zeros -2, 3, and -6 is f(x) = \(x^{3}\) + 5\(x^{2}\) - 42.
The zeros of a polynomial function are the values of x for which the function evaluates to zero. Since we are given the zeros -2, 3, and -6, we can express the polynomial function as the product of factors corresponding to each zero.
To find the polynomial, we use the fact that if a number is a zero of a polynomial, then the corresponding factor is equal to zero. So, for the zeros -2, 3, and -6, we have the factors (x + 2), (x - 3), and (x + 6), respectively.
Multiplying these factors together, we get:
f(x) = (x + 2)(x - 3)(x + 6)
Expanding the expression further:
f(x) = (\(x^{2}\) - x - 6)(x + 6)
= (\(x^{2}\) - x - 6)(x) + (\(x^{2}\) - x - 6)(6)
= \(x^{3}\) - \(x^{2}\) - 6x + 6\(x^{2}\) - 6x - 36
= \(x^{3}\) + 5\(x^{2}\) - 42
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Explain how to find the value of y that would make segment OP parallel to segment LN.
Answers
Answer:
\(y = 36\)
Step-by-step explanation:
Given; the triangle above
Required
Find y
This question falls under the topic/subtopic similar triangles where we need to make comparison between similar sides'
But first, it should be noted that triangle MOP is similar to triangle MLN
This implies that
Side MP is similar to MN
Side MO is similar to ML
Mathematically; This can be represented as follows;
\(\frac{MP}{MN} = \frac{MO}{ML}\)
Where MP =y; MN = y + 18; MO = 28; ML = 28 + 14
Substitute these values in the above expression
\(\frac{y}{y+18} = \frac{28}{28+14}\)
\(\frac{y}{y+18} = \frac{28}{42}\)
Multiply both sides by 42
\(42 * \frac{y}{y+18} = \frac{28}{42} * 42\)
\(\frac{42y}{y+18} = \frac{28*42}{42}\)
\(\frac{42y}{y+18} = 28\)
Multiply both sides by y + 18
\((y+18)*\frac{42y}{y+18} = 28 * (y+18)\)
\(42y = 28 * (y+18)\)
Open Bracket
\(42y = 28 * y+28 * 18\)
\(42y = 28y+504\)
Subtract 28y from both sides
\(42y - 28y = 28y - 28y +504\)
\(14y = 504\)
Divide both sides by 14
\(\frac{14y}{14} = \frac{504}{14}\)
\(y = \frac{504}{14}\)
\(y = 36\)
Answer:
y= 36
Step-by-step explanation:
also ily, no homo
There is a line that includes the point (4,7) and has a slope of –1/6. What is its equation in point-slope form?
Answers
The point- slope form of the line is y-7 = -0.17(x-4).
What is line?
A line is an one-dimensional figure. It has length but no width. A line can be made of a set of points which is extended in opposite directions to infinity. There are straight line, horizontal, vertical lines or may be parallel lines perpendicular lines etc.
There is a line that includes the point (4,7) and has a slope of –1/6.
Any line in point - slope form can be written as
y - y₁= m(x -x₁) -------(1)
where,
y= y coordinate of second point
y₁ = y coordinate of first point
m= slope of the line
x= x coordinate of second point
x₁ = x coordinate of first point
In the given problem (x₁ , y₁) = (4, 7) and m= -1/6
Putting all these values in equation (1) we get,
y-7= (-1/6) (x- 4)
⇒ y-7 = -0.17(x-4)
Hence, the point- slope form of the line is y-7 = -0.17(x-4).
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Several customers order small fruit baskets filled with apples, bananas, and oranges. Let a represent the price per pound of apples, b represent the price per pound of bananas, and c represent the price per pound of oranges. The system represents the number of pounds of each type of fruit and the total price of each fruit basket. How much per pound does each type of fruit cost
Answers
The missing equations are;
a + 2b + 3c = 12
3a + 2b + 5c = 22
2a + 4b + 4c = 18
Answer:
a = 2
b = 0.5
c = 3
Step-by-step explanation:
We are told that;
a represent the price per pound of apples
b represent the price per pound of bananas
c represent the price per pound of oranges
The equations system is;
a + 2b + 3c = 12 - - - (eq 1)
3a + 2b + 5c = 22 - - - (eq 2)
2a + 4b + 4c = 18 - - - (eq 3)
Divide through equation 3 by 2 to get
a + 2b + 2c = 9 - - - (eq 4)
Subtract eq 4 from eq 1 to get;
3c - 2c = 12 - 9
c = 3
Put 3 for c in eq(2) and eq 4;
3a + 2b + 5c = 22
3a + 2b + 5(3) = 22
3a + 2b + 15 = 22
3a + 2b = 7 - - - (5)
a + 2b + 2(3) = 9
a + 2b + 6 = 9
a + 2b = 3 - - - (eq 6)
Subtract eq 6 from eq 5 to get;
2a = 7 - 3
2a = 4
a = 2
Put 2 for a in eq 6;
2 + 2b = 3
2b = 3 - 2
2b = 1
b = 1/2
b = 0.5
Answer:
It's A
Step-by-step explanation:
edge 2021
#3:Find the slope of the line below
Answers
Answer:
-5/2
Step-by-step explanation:
with slope always remember rise over run, vertically, the dots are 5 points away from each other so that is your rise. Horizontally they are only 2 away so that's your run. the answer is negative because the slope goes down to the right. If it went up to the right then the slope would be positive.
-5/2
Explanation:
The formula for slope is rise/run. If you start at the point (1, -2), you can go up 5 and left 2 to get to the other point.
Since the line goes downwards, the slope must be a negative. This leaves you with -5/2.
Hope this helps!
2.45 convert the following unsigned binary numbers to hexadecimal. a. 1101 0001 1010 1111 b. 001 1111 c. 1 d. 1110 1101 1011 0010
Answers
a. 1101 0001 1010 1111 --> D1AF, b. 001 1111 --> 1F, c. 1 --> 1, d. 1110 1101 1011 0010 --> EDB2.
What is the hexadecimal representation of the given binary numbers?
Converting binary numbers to hexadecimal involves grouping the binary digits into sets of four, starting from the rightmost digit. Each group is then converted to its corresponding hexadecimal digit.
In the first step, we convert the binary numbers to hexadecimal as follows:
a. 1101 0001 1010 1111 --> D1AF
b. 001 1111 --> 1F
c. 1 --> 1
d. 1110 1101 1011 0010 --> EDB2
In binary, each digit represents a power of 2, while in hexadecimal, each digit represents a power of 16.
The conversion simplifies the representation and allows for easier understanding and manipulation of binary numbers.
Learn more about binary-to-hexadecimal conversion.
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It is a chilly January day. When you get up it is only 5°F outside. By noon, the temperature had risen 7°F. By 10pm, it had dropped 15°F. What is the temperature outside when you go to bed at 10pm?
Answers
Answer:-3
Step-by-step explanation:
its just adding and subtracting
Answer:
-2ºf
Step-by-step explanation:
the temp is 5ºf outside in the morning, then it went up 7ºf more, so 5+7= 13
after that, you would then do 13-15 which would be -2
so by 10pm it would be -2ºf