The sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation.
To prove that for all n ≥ 0, the sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation, we can use mathematical induction.
Base case:
For n = 0, we have h0 = 5/3 which is equal to the given initial value.
For n = 1, we have h1 = 11/3 which is also equal to the given initial value.
Inductive step:
Assume that the recurrence relation holds for some k ≥ 1, i.e., hk = 3hk-1 - 4hk-2 + 6k.
We want to show that it also holds for k+1, i.e., h(k+1) = 3h(k+1)-1 - 4h(k+1)-2 + 6(k+1).
Using the recurrence relation for hk, we have:
hk+1 = 3hk - 4hk-1 + 6k+3 (by substituting k+1 for n in the given recurrence relation)
Similarly, we have:
hk = 3hk-1 - 4hk-2 + 6k (by assumption)
hk-1 = 3hk-2 - 4hk-3 + 6(k-1) (by assumption)
Substituting these values into the expression for hk+1, we get:
hk+1 = 3(3hk-1 - 4hk-2 + 6k) - 4(3hk-2 - 4hk-3 + 6(k-1)) + 6(k+1)
Simplifying the expression, we get:
hk+1 = 9hk-1 - 12hk-2 + 18k - 12hk-2 + 16hk-3 - 24(k-1) + 6(k+1)
hk+1 = 9hk-1 + 4hk-3 - 12hk-2 - 6(k-1) + 6(k+1)
hk+1 = 3(3hk-1 - 4hk-2 + 6k+1) - 4(3hk-2 - 4hk-3 + 6k) + 6(k+1)
hk+1 = 3h(k+1)-1 - 4h(k+1)-2 + 6(k+1)
This shows that the recurrence relation holds for all n ≥ 0 by mathematical induction, and hence the sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation.
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A management consulting firm recommends that the ratio of middle-management salaries to management trainee salaries be 8:7. Using this recommendation, what is the annual middle-management salary if the annual management trainee salary is $17,000? (Round your answer to the nearest dollar.)
The management salary is $19,429
What is ratio?
Ratio is the quantitative relationship between two numbers which shows the number of times an amount or a number is contained within the other amount or number.
In this case, a ratio of 8:7 shows the number of times the management salaries is contained within the management trainee salary
The management salary has a value of 8 whereas the management trainee salary has a value of 7 as shown below:
salary ratio=management salary/management trainee salary
salary ratio=8/7
management salary=unknown(assume it is X)
management trainee salary=$17,000
8/7=X/$17,000
cross multiply
8*$17,000=7*X
X=8*$17,000/7
X=$19,429
The fact that annual management trainee salary is $17,000 means that the management salary is $19,429 annually.
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Please help with these problems having trouble!!
The function f(x) is vertically compressed to form g(x) while the function f(x) is vertically compressed and then reflected across the x-axis to form h(x)
How to compare both functions?The functions are given as
f(x) =x^2
g(x) =3x^2
h(x) = -3x^2
Substitute f(x) =x^2 in g(x) =3x^2 and h(x) = -3x^2
g(x) =3f(x)
h(x) = -3f(x)
This means that the function f(x) is vertically compressed to form g(x)
Also, the function f(x) is vertically compressed and then reflected across the x-axis to form h(x)
See attachment for the functions g(x) and h(x)
Also, functions f(x) and g(x) have the same domain and range
While functions f(x) and h(x) have the same domain but different range
The complete table is:
x -2 -1 0 1 2
g(x) 12 3 0 3 12
h(x) -12 -3 0 -3 -12
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a) There are 12 different varieties of flowers. How many different kinds of bouquets can be made using 5 different kinds of flowers
Total of 1820 different kinds of banquets can be made.
What is combination in mathematics?Combinations are mathematical procedures that count how many different ways a collection of items could be arranged when the order in which they are chosen is immaterial. Combos' components can be chosen in any hierarchy.
[tex]$C_{12}(16)=\left(\begin{array}{l}16 \\ 12\end{array}\right)=\frac{16 !}{12 !(16-12) !}=\frac{16 \cdot 15 \cdot 14\cdot13}{4\cdot3 \cdot 2 \cdot 1}=1820[/tex]
[tex]$n=5$ $k=12$\\$x=\left(\begin{array}{c}n+k-1 \\ k\end{array}\right)=\left(\begin{array}{c}5+12-1 \\ 12\end{array}\right)=1820$[/tex]
Total of 1820 different kinds of banquets can be made
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4. (06.02 MC)
Write an equation of a line that passes through the point (7,3) and is parallel to the line y =
y=-3x+3.0
x + 3. (5 points)
Answer:
Step-by-step explanation:
Givens
Line: y = 3x + 3
Point: (7,3)
Discussion
The easiest variation on line problems is one like this one. The reason is that all you need do is read the slope of the given line.
m = 3 of the given line. Also the slope of the new line.
What you have so far is
y = 3x + b
The point is used to find b -- the y intercept.
x = 7
y = 3
You are not given this, but what you want to think is that when x = 7, y = 3
Substitute into the new equation
3 = 3*7 + b
3 = 21 + b Subtract 21 from both sides.
3 - 21 = 21-21 + b
- 18 = b
Answer
y = 3x - 18
Answer:
-2/3x+23/3
Step-by-step explanation:
y= mx+c
where
m is the slope c is the y-intercept.
1-getting the slope:
we are given that the line we are looking for is parallel to the line
y=-2/3x+3
this means that the slope of the two lines are equal . comparing the general formula with the given line , we would find that the slope of the given lines -2/3.this means that the slope of the lines we are looking for is also -2/3
The equation of the line now becomes:y=-2/3x+c
2-getting the value of c
to get the value of c, we will use a point that belongs to the line (7,3),substitute in the equation and solve for c as follows:
y=-2/3x+c
3=(-2/3)(7)+c
3=-14/3+c
c=3+14/3
c=23/3
Based on the above, the equation of the line we are looking for is:
y=-2/3x+23/3
Volume=
Help me please!! Asap thanks so much
The volume of the oblique cylinder is 6867.18 cubic units
How to determine the volume?The given parameters are:
Height = 27
Radius = 9
The volume is calculated as:
[tex]V = \pi r^2 h[/tex]
So, we have:
[tex]V = 3.14 * 9^2 * 27[/tex]
Evaluate
V = 6867.18
Hence, the volume of the oblique cylinder is 6867.18 cubic units
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The length of tr is 17 units. what are the lengths of sv and qt? sv = units qt = units
Applying the properties of kite, the measures are: SV = 41 units; QT = 21 units.
What is a kite?A kite is a quadrilateral that has 2 pairs of congruent adjacent sides, and has two diagonals where the longest one bisects the shorter one and divides it into equal segments.
The diagram given is a kite. Where TR = 17 units. Thus we would solve for x using the equation as shown below:
TR = RV [congruent adjacent sides of the kite]
Plug in the values
17 = 3x + 2
17 - 2 = 3x
15 = 3x
15/3 = x
x = 5
SV = TS = 9x - 4 [congruent adjacent sides of the kite]
Plug in the value of x
SV = 9(5) - 4
SV = 41 units
The length of segment QT is 41 units.
QT = QV = 4x + 1 [congruent adjacent sides of the kite]
Plug in the value of x
QT = 4(5) + 1
QT = 21 units
The length of segment QT is: 21 units.
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Answer:
SV: 41
QT:21
Step-by-step explanation:
If the growth rate of bacteria at any time t is proportional to the number present at t and triples in 1 week
If the growth rate of bacteria at any time is proportional to the number of bacteria present at t then the population after 20 weeks will be 403.42[tex]x_{0}[/tex] in which [tex]x_{0}[/tex] is the initial population.
Given that the growth rate of bacteria at any time t is proportional to the number present at t and triples in 1 week.
We are required to find the number of bacteria present after 10 weeks.
let the number of bacteria present at t is x.
So,
dx/dt∝x
dx/dt=kx
1/x dx=k dt
Now integrate both sides.
[tex]\int\limits {1/x} \, dx[/tex]=[tex]\int\limits{k} \, dt[/tex]
log x=kt+log c----------1
Put t=0
log [tex]x_{0}[/tex]=0 +log c ([tex]x_{0}[/tex] shows the population in beginning)
Cancelling log from both sides.
c=[tex]x_{0}[/tex]
So put c=[tex]x_{0}[/tex] in 1
log x=kt+log [tex]x_{0}[/tex]
log x=log [tex]e^{kt}[/tex]+log [tex]x_{0}[/tex]
log x=log [tex]e^{kt}x_{0}[/tex]
x=[tex]e^{kt}x_{0}[/tex]
We have been given that the population triples in a week so we have to put the value of x=2[tex]x_{0}[/tex] and t=1 to get the value of k.
2[tex]x_{0}[/tex]=[tex]e^{k} x_{0}[/tex]
2=[tex]e^{k}[/tex]
log 2=k
We have to now put the value of t=20 and k=log 2 ,to get the population after 20 weeks.
x=[tex]e^{20log 2}[/tex][tex]x_{0}[/tex]
x=[tex]e^{0.30*2}[/tex][tex]x_{0}[/tex]
x=[tex]e^{6}x_{0}[/tex]
x=403.42[tex]x_{0}[/tex]
If the growth rate of bacteria at any time is proportional to the number of bacteria present at t then the population after 20 weeks will be 403.42[tex]x_{0}[/tex] in which [tex]x_{0}[/tex] is the initial population.
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The given question is incomplete as the question incudes the following:
Calculate the population after 20 weeks.
calculate problems one and two.
Answer:
Question 1) 2/3
Question 2) -3
Step-by-step explanation:
Question 1)
The formula of a slope is (increase in y)/(increase in x), so in the graph, all we need to do is find two precise points and look at the distance difference.
1. The two points that I've found were (0,3) and (3,5). The change in y is 3 --> 5, which is 2. The change in x is 0 --> 3, which is 3.
2. The slope will be 2/3.
Question 2)
In this question, we can consider n as x and f(n) as y.
1. The two points I chose were (4, -31) and (7,-40). The change in y is -31 --> -40, which is -9. The change in x is 4 --> 7, which is 3.
2. The slope will be -9/3.
3. -9/3 can be simplified to -3 since we can divide both the numerator and the denominator by 3.
4. The final slope will be -3.
.
7-9p≥7 or 2p - 9> 11
Answer:
2p-9>
Step-by-step explanation:
Select the correct answer from the drop-down menu.
Jaden and his family are using a map to drive down to Napa Valley. On the map, the distance is given by the equation c = 1.3m, where c is the number of centimeters on the map and m is the corresponding real-world distance in miles.
If they cover a distance of 18 miles, that would mean a distance of
centimeters on the map.
Answer:
13.846153846
Step-by-step explanation:
Its easy Just 18:1.3=13.846153846
Which of the following are factor pairs for 12?
1. 1 and 12
2. 2 and 4
3. 2 and 6
3. 3 and 4
4. 3 and 5
NEED ANSWER QUICK PLEASE!!
Answer:
FACTOR pairs are a set of two integers that when multiplied together gives a particular product.FACTOR pairs of 12 are :1 and 12 because 1 x 12 = 122 and 6 because 2 x 6 =123 and 4 because 3 x 4 = 12Hope you understandHope this helps.Good luck ✅.3 + 2 x 7 - 30(^2)
help me
HELP ME
ANY GENIUS OUT THERE>
Answer:
angle x equals 25
Step-by-step explanation:
130 + 25 = 155
180 - 155 = 25
Triangles always add up to 180°
Answer:
it is so easy the answer is 25
Step-by-step explanation:
The total sum of the traingle is 180 degree. So frist add both the numbers in the traingle and then subtract it by 180 degree
Determine S9 of the series where a=6 and r=2
Hi
arithmetic serie mean that to change from a rank to an other, you add "r"
So : if 6 is S1, and 2 is pace,
then you need 8 hops to go from S1 to S9.
so you 'r going to add the reason 8 times.
so : S9 = 6 + 8 * 2 = 6 + 16 = 22
conclusion : S9 = 22
PLEASE HELP IM SO STUCK
Answer:
Slope = 7/4
Step-by-step explanation:
You are given coordinate values and a formula that tells you what to do with them. All you need to do is put the values in the formula and do the arithmetic.
PointsYou are given points (2, -3) and (6, 4). The slope formula refers to the two points as (x1, y1) and (x2, y2). It does not matter which is which.
If we use the points in the same order they are given, then we can see that ...
x1 = 2y1 = -3x2 = 6y2 = 4FormulaUsing these values in the given formula, we find the slope to be ...
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{4-(-3)}{6-2}=\dfrac{7}{4}[/tex]
The slope of the line through the given points is 7/4.
Find a polynomial function of degree 7 with a leading coefficient of 1 and with - 3 as a zero of multiplicity 3,0 as a zero of multiplicity 3, and 3 as a zero of multiplicity 1.
The least polynomial in standard form is defined by the function f(x) = x⁷ + 6 · x⁶ - 54 · x⁴ - 81 · x³.
How to determine the least polynomial given a set of roots and a leading coefficient
Polynomials can be expressed as a product of binomials of the form (x - r) multiplied by a leading coefficient. The least polynomial contain the number of roots presented in statement, whose factor form is shown below:
f(x) = 1 · (x + 3)³ · x³ · (x - 3)
f(x) = (x + 3)³ · (x⁴ - 3 · x³)
f(x) = (x³ + 9 · x² + 27 · x + 27) · (x⁴ - 3 · x³)
f(x) = x⁷ + 9 · x⁶ + 27 · x⁵ + 27 · x⁴ - 3 · x⁶ - 27 · x⁵ - 81 · x⁴ - 81 · x³
f(x) = x⁷ + 6 · x⁶ - 54 · x⁴ - 81 · x³
The least polynomial in standard form is defined by the function f(x) = x⁷ + 6 · x⁶ - 54 · x⁴ - 81 · x³.
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what is the solution to the system of equation -4+3y=-3
Answer:
[tex]\huge\boxed{\sf y = 1/3}[/tex]
Step-by-step explanation:
Given equation:-4 + 3y = -3
Add 4 to both sides
3y = -3 + 4
3y = 1
Divide 3 to both sides
y = 1/3[tex]\rule[225]{225}{2}[/tex]
. A special safe lets you choose from 9 symbols for a 3-symbol-long pass code. You may enter a symbol any number of times. How many potential pass codes are there
The question is an illustration of combination and there are 729 potential pass codes available
How to determine the number of potential pass codes?The given parameters are
Symbols available = 9
Length of pass code = 3
From the question, we understand that a symbol may be entered any number of times.
This means that each of the 9 available symbols can be used three times
So, the number of potential pass codes is
Passcodes = 9 * 9 * 9
Evaluate the product
Passcodes = 729
Hence, there are 729 potential pass codes available
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Jim is driving to Denver. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes). Jim has 61 miles to his destination after 41 minutes of driving, and he has 42.3 miles to his destination after 63 minutes of driving. How many miles will he have to his destination after 81 minutes of driving
Jim has to travel 26.82 miles more to reach Denver.
What is the equation of a line?
Let the first point be = (x₁ , y₁)
Let the second point be = (x₂ , y₂)
m(slope) = (y₂ - y₁) / (x₂ - x₁)
Equation of line = (y - y₁) = m(x - x₁)
Let the no. of miles be plotted on X-axis
Let the driving time be plotted on Y-axis
First point = (61 , 41)
Second point = (42.3 , 63)
Slope of the linear function = (63 - 41)/(42.3 - 61) = -1.17
Equation of the line = (y - 41) = -1.17(x - 61)
Given time of driving = 81 min
Number of miles = (81 - 41) = -1.17(x - 61)
40/-1.17 = x - 61
-34.18 + 61 = x
x = 26.82
Hence, Jim has to travel 26.82 miles more to reach Denver.
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Find the area of a circle with diameter, d = 3.3m. give your answer rounded to 1 dp.
Answer:
Step-by-step explanation:
the area of a circle is πr²
r being the radius, which is equal to half the diameter
3.3/2= 1.65
π(1.65)²=8.6m
Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Yeony2202, tysm for the help!
These two should be the points as then 2 points will remain on both sides
Slope
m=(7+2)/6+6m=9/12m=3/4Equation in point slope form
y+2=3/4(x+6)y=3/4x+6-2y=3/4x+4When x=-15
y=3/4(-15)+4y=-45/4+4y=(-45+16)/4y=--29/4Just above 7
WILL MAKE BRAINLIEST!! If m
Answer:
98
Step-by-step explanation:
m<KJU + m<UJI = m<KJI
this equation will help us find the value of x therefore the value of m<KJU
Rewrite the equation using the expressions
104 + x + x + 76 = 168 add like terms
180 + 2x = 168 subtract 180 from both sides
2x = -12 divide both sides by 2
x = -6 to find the value of m<KJU replace x with -6
104 + (-6) = 98
Ms. johnson owns a spice shop. she combines 5 pounds of dried rose hips and 1 pound of five-spice powder to make a new seasoning. how much should she charge for 1 ounce of the new seasoning if she can sell the dried rose hips for $8 per pound and the five-spice powder for $11 per ounce?
Given the 6 pound weight of the new seasoning having ingredients that cost $8, and $11, the amount she should charge for 1 ounce is $0.53125
Which method can be used to calculate the price of the new seasoning?The ingredients in the new seasoning are;
5 pounds of dried rose hips + 1 pounds of five-spice powder
Cost of the ingredients are;
1 pound of dried rose hips = $8
1 pound of five-spice powder = $11
The cost of combined spice, C, is therefore;
C = $8 × 5 + $11 × 1 = $51
The weight of the new combined seasoning = (5 + 1) pounds = 6 pounds
Cost of 1 pound of the new seasoning is therefore;
[tex]1 \: lb = \frac{51}{6} = 8.5[/tex]
1 lb of the new seasoning = $8.51 lb = 16 ouncesTherefore;
[tex]1 \: ounce = \frac{8.5}{16} = \mathbf{0.53125}[/tex]
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Answer: $2.25
Step-by-step explanation:
5 lbs = 80 oz, so the mixture is worth
80(8/16) + 16(11) = 96x
x = $2.25/oz
Hope this helps!!
Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Yeony2022, tysm for the help!
Answer:
d: y=5x
Step-by-step explanation:
Since we know a few points: (2,10), (3,15), (4,20), we can use two of them to first find the slope. The equation for the slope is (Increase in y)/(increase in x).
1. Slope between (2,10) and (3,15) is 5/1 which is 5. So the slope will be 5.
2. With the slope, we know that our equation will be: y=5x+b. To find b, we can replace x and y with another point. 20 = 5(4)+b. --> 20 = 20 +b. Now we know that b equals 0.
Our final equation will be y = 5x.
RECHECKING!
I'll recheck whether my answer is right or not using Desmos (A online graphing calculator).
I added an image down below, and I drew the points in blue. This shows that since the graph crosses all three points, it is correct.
WHY y=1/5x IS NOT CORRECT (based on your comments)
If you look at the points, the relationship between x and y is not 1/5. If you multiply something from x, it should become 10.
--> 2 times __ = 10
--> 3 times __ = 15
--> 4 times __ = 20
And clearly, __ will be 5. If it becomes 1/5, we won't get the right number of y.
This is why y=1/5 is not correct.
Hope this helps!
A family needs to build fencing around their rectangular home and square swimming pool, depicted below.
A rectangle labeled home has its right side labeled (2 + 5 x) yards and its bottom side labeled (3 + 10 x) yards. The square, labeled pool, is smaller than the rectangle. One of its sides is labeled (2x) yards.
The total amount of fencing they need can be written as x + 10.
If x = 5;
The family needs yards of fencing for their home.
The family needs yards of fencing for their pool.
The total amount of fencing they need = 38x + 10.
Perimeter for the rectangular fencing of the home = 160 yards
Perimeter for the square fencing of the pool = 40 yards
What is the Perimeter of a Rectangle and a Square?Perimeter of a rectangle = 2(length + width)
Perimeter of a square = 4(side length).
Perimeter for the rectangular fencing of the home = 2(3 + 10x + 2 + 5x) = 2(15x + 5) = 30x + 10
Perimeter for the square fencing of the pool = 4(2x) = 8x
The total amount of fencing they need = 30x + 10 + 8x = 38x + 10.
If x = 5, then:
Perimeter for the rectangular fencing of the home = 30x + 10 = 30(5) + 10 = 160 yards
Perimeter for the square fencing of the pool = 4(2x) = 8x = 8(5) = 40 yards.
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Answer:
38
160
40
Step-by-step explanation:
You are the CEO of a corporation that makes computer hard drives. The company made $50 million dollars in annual profit this year, after struggling for the 5 previous years. About 6,000 people are working for your company and about 80% of them are directly associated with the manufacturing plant. Workers at the plant have not received a salary raise for the past 5 years, and the workers union is demanding a salary raise this year. The company has an annual payroll of about $500 million dollars, with about 70% of it for the workers in the plant. Factory managers are asking for $25 million dollars to upgrade the equipment and improve the efficiency of the plant. White-collar workers in the head office claim that it is their strategy that turned the factory around and put it on the path to profit. Due to this reason, they think they deserve to get a big bonus. How do you allocate the profits to meet all these demands? Justify your decision.
When it comes to resource allocation, it's not a one sentence answer. The explanation for how to go about this is given below. But first, lets look at what Profit Allocation means.
What is profit allocation?This simply means the process of distributing profit to various areas of the business such that all parts are (human and non-human components) are satisfied and operating at equilibrium.
So how does one allocate resources?There are five steps to aligning your budget and resource demands when planning for support resource allocation.
Recognize your financial flow.Outline your present staffing and resource requirements.Examine your present budget.Create firm growth objectives.Align your budget with your objectives.Learn more about profit allocation at;
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Find the equation of the line perpendicular to 2x+3y-6=0 and passing through (-1,1)
Answer: [tex]y-1=\frac{3}{2}(x+1)[/tex]
Step-by-step explanation:
[tex]2x+3y-6=0\\\\3y=-2x+6\\\\y=-\frac{2}{3}x+2[/tex]
So, the slope of the given line is -2/3, meaning the slope of the line we want to find is 3/2.
Substituting into point-slope form, we get the equation of the line is
[tex]y-1=\frac{3}{2}(x+1)[/tex]
A school offers three subjects: math, art and science. at least 80% of students study both math and art. at least 80% of students study both math and science. prove that at least 80% of students who study both art and science, also study math.
The proof to show that at least 80% of students who study both art and science, also study math is illustrated below.
How to illustrate the proof?From the information given, it was stated that at least 80% of students study both math and art and that at least 80% of students study both math and science.
Therefore, about (100 - 80) 20% doesn't study the combination of both subjects.
Here, those students who study both art and science, and also study math will be:
= 100% - 20%
= 80%
Therefore, at least 80% study the subjects.
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Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the help!
Answer:
m=15
Step-by-step explanation:
Oh, I think I answered this question recently, but it's m=15c
The explanation for the question was written in my other response.
Answer:
d. m = 15c
Step-by-step explanation:
Directly proportional means as one amount increases, another amount increases at the same rate.
Proportionality can be used to set up an equation.
Given variables:
m = amount of money earned (in dollars)c = number of cars washedIf m is directly proportional to c then: [tex]m \propto c[/tex]
Convert this to an equation by using a constant of proportionality, k:
[tex]\implies m=kc[/tex]
We are told that Paul washed 8 cars and earned $120.
Substitute these values into the found equation and solve for k:
[tex]\implies 120=8k[/tex]
[tex]\implies k=\dfrac{120}{8}[/tex]
[tex]\implies k=15[/tex]
Finally, substitute the found constant of proportionality (k) into the equation:
[tex]\implies m=15c[/tex]
The leaning Tower of Pisa was completed in 1372 and makes 86 degree angle with the ground. The tower is 50 meters tall, measured vertically from the ground to the highest point. If you were to climb to the top then accidentally drop your keys, where would you start looking for them? question 5
Answer:
you would look for them 3.5 meters away from the base
Step-by-step explanation:
so u do tan 86 =50/x
and u solve from there