Notice that the difference in the absolute values of consecutive coefficients is constant:
|-7| - 1 = 6
13 - |-7| = 6
|-19| - 13 = 6
and so on. This means the coefficients in the given series
[tex]\displaystyle \sum_{i=1}^\infty c_i a^{i-1} = \sum_{i=1}^\infty |c_i| (-a)^{i-1} = 1 - 7a + 13a^2 - 19a^3 + \cdots[/tex]
occur in arithmetic progression; in particular, we have first value [tex]c_1 = 1[/tex] and for [tex]n>1[/tex], [tex]|c_i|=|c_{i-1}|+6[/tex]. Solving this recurrence, we end up with
[tex]|c_i| = |c_1| + 6(i-1) \implies |c_i| = 6i - 5[/tex]
So, the sum to [tex]n[/tex] terms of this series is
[tex]\displaystyle \sum_{i=1}^n (6i-5) (-a)^{i-1} = 6 \underbrace{\sum_{i=1}^n i (-a)^{i-1}}_{S'} - 5 \underbrace{\sum_{i=1}^n (-a)^{i-1}}_S[/tex]
The second sum [tex]S[/tex] is a standard geometric series, which is easy to compute:
[tex]S = 1 - a + a^2 - a^3 + \cdots + (-a)^{n-1}[/tex]
Multiply both sides by [tex]-a[/tex] :
[tex]-aS = -a + a^2 - a^3 + a^4 - \cdots + (-a)^n[/tex]
Subtract this from [tex]S[/tex] to eliminate the intermediate terms to end up with
[tex]S - (-aS) = 1 - (-a)^n \implies (1-(-a)) S = 1 - (-a)^n \implies S = \dfrac{1 - (-a)^n}{1 + a}[/tex]
The first sum [tex]S'[/tex] can be handled with simple algebraic manipulation.
[tex]S' = \displaystyle \sum_{i=1}^n i (-a)^{i-1}[/tex]
[tex]\displaystyle S' = \sum_{i=0}^{n-1} (i+1) (-a)^i[/tex]
[tex]\displaystyle S' = \sum_{i=0}^{n-1} i (-a)^i + \sum_{i=0}^{n-1} (-a)^i[/tex]
[tex]\displaystyle S' = \sum_{i=1}^{n-1} i (-a)^i + \sum_{i=1}^n (-a)^{i-1}[/tex]
[tex]\displaystyle S' = \sum_{i=1}^n i (-a)^i - n (-a)^n + S[/tex]
[tex]\displaystyle S' = -a \sum_{i=1}^n i (-a)^{i-1} - n (-a)^n + S[/tex]
[tex]\displaystyle S' = -a S' - n (-a)^n + \dfrac{1 - (-a)^n}{1 + a}[/tex]
[tex]\displaystyle (1 + a) S' = \dfrac{1 - (-a)^n - n (1 + a) (-a)^n}{1 + a}[/tex]
[tex]\displaystyle S' = \dfrac{1 - (n+1)(-a)^n + n (-a)^{n+1}}{(1+a)^2}[/tex]
Putting everything together, we have
[tex]\displaystyle \sum_{i=1}^n (6i-5) (-a)^{i-1} = 6 S' - 5 S[/tex]
[tex]\displaystyle \sum_{i=1}^n (6i-5) (-a)^{i-1} = 6 \dfrac{1 - (n+1)(-a)^n + n (-a)^{n+1}}{(1+a)^2} - 5 \dfrac{1 - (-a)^n}{1 + a}[/tex]
[tex]\displaystyle \sum_{i=1}^n (6i-5) (-a)^{i-1} =\boxed{\dfrac{1 - 5a - (6n+1) (-a)^n + (6n-5) (-a)^{n+1}}{(1+a)^2}}[/tex]
find the value of x. round to the nearest tenth
Answer:
*79.40*
Step-by-step explanation:
using sine rule..
a/sinA = b/sinB
42/sin19= x/sin142
x= sin142*42/sin19
= 79.42356..
to the nearest tenth = 79.40.
Answer:
Step-by-step explanation:
a/sinA = b/sinB
(42/sin19) = (x/sin142)
x = (sin142 *42 / sin19)
Round to the nearest tenth -> 79.4
3. Measure the length of each leg and
the hypotenuse of this triangle:
AD =
units
AE=
units
DE=
units
Correct!
Check
The length of each leg and the hypotenuse of this triangle will be:
AD = 2 units.
AE = 2 units.
DE = 2.8 units
How to compute the hypotenuse?From the information given and the attached diagram, the value or AD and AE are 2 units respectively.
The value of DE will be calculated as the square root of 2² + 2². This will be:
= 2² + 2² = 8
= ✓8 = 2.8
Therefore, the correct options are 2, 2, and 2.8 units.
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Find the inverse of the function.
y=x+9
Answer: [tex]y=x-9[/tex]
Step-by-step explanation:
To find the inverse of a function you must first swap the places of x and y
[tex]x=y+9[/tex]
Now solve for y
Step 1: Flip the equation.
[tex]y+9=x[/tex]
Step 2: Subtract 9 from both sides.
[tex]y+9-9=x-9\\y=x-9[/tex]
As an inventory manager, you must decide on the order quantity for an item. Its annual demand is 679 units. Ordering costs are $7 each time an order is placed, and the holding cost is 10% of the unit cost. Your supplier provided the following price schedule. Quantity Price per Unit 1 - 100 $5.65 101 - 350 $4.95 351 or more $4.55 What ordering-quantity policy do you recommend
The ordering-quantity that should be recommended will be 117 units.
How to calculate the order quantity?The following can be deduced from the information given:
Demand = 679
Ordering cost = 7
Holding cost = 10%
Ordering quantity will be:
= ✓(2 × Demand × Ordering cost / Holding cost)
= ✓(2 × 679 × 7/0.1)
= ✓13580
= 116.53
= 117
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Analyze the diagram below and complete the instructions. Find the value of x and the value of y
The value of the variable x and y will be 15 and 5√3. Then the correct option is D.
The complete question is attached below.
What is trigonometry?The connection between the lengths and angles of a triangular shape is the subject of trigonometry.
The right triangle is shown.
Then the value of x will be
x = 10√3 × sin 60°
x = 15
Then the value of y will be
y = 10√3 × cos 60°
y = 5√3
Then the correct option is D.
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Find the interval in which the function f(x) = |x + 4| is increasing.
[tex]f { }^{ - } (x) = \frac{2(x + 4)}{2 |x + 4| {}^{} } = \frac{x + 4}{ |x + 4| } [/tex]
[tex] |x + 4| \: is \: always \: positive[/tex]
[tex]solve \: x + 4 > 0[/tex]
[tex]x > - 4[/tex]
[tex]f(x) \: is \: incresing \: from \\[/tex]
]-4,∞[Considering only the values of β for which (1+cosβ)(1−cosβ)sinβ is defined, which of the following expressions is equivalent to (1+cosβ)(1−cosβ)sinβ?
Select the correct answer below:
sinβ
sin3β
secβ
1
sec2β
The following expressions (1+cosβ)(1−cosβ)sinβ is equivalent to sin³β
What are Trigonometric Ratios ?In a Right angled triangle , trigonometric ratios can be used to determine the value of angles and sides of the triangle.
The trigonometric expression given in the question is
(1+cosβ)(1−cosβ)sinβ
(a+b)(a-b) = a² - b²
( 1 - cos²β)sinβ
By the trigonometric Identity
1-cos²β = sin² β
sin² β x sin β
sin³β
Therefore Option B is the correct answer.
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Rationalize the Denominator
2i/i-2
Answer:
2-4i/5
Step-by-step explanation:
[tex]\frac{2i}{i-2} *\frac{i+2}{i+2}\\=\frac{2i(i+2)}{i^2-4}\\= \frac{2i(i+2)}{-1-4}\\ =\frac{2i(i+2)}{-5} \\ = \frac{-2 + 4i}{-5}\\ =( 2 - 4i )/ 5[/tex]
Answer:
[tex]\frac{2i}{i-2}=\frac{2-4i}{5}[/tex]
Step-by-step explanation:
[tex]\frac{2i}{i-2}[/tex] (Given)[tex]=\frac{2i}{i-2}\times \frac{i+2}{i+2}[/tex] [Multiply numerator and denominator both by (i + 2)][tex]=\frac{2i(i+2)}{(i-2)(i+2)}[/tex][tex]=\frac{2i^2+4i)}{i^2-2^2}[/tex][tex]=\frac{2(-1)+4i)}{-1-4}[/tex][tex]=\frac{-2+4i)}{-5}[/tex][tex]=\frac{2-4i)}{5}[/tex][tex]\implies \frac{2i}{i-2}=\frac{2-4i}{5}[/tex]Which expression is equivalent to the given expression? 4lh x + lh 3 - lh x
ON ECRIT SUR LES DUN DE EQUILIBRE CHACUNE DES LETTRES AMURES
Answer:
oh.
Step-by-step explanation:
f(x)=x2 what is g(x)
Answer:
-x^2-2
Step-by-step explanation:
So by looking at the two curves it appears that g(x) is simply a reflection of f(x) over the x-axis and then shifted 2 units down. So to reflect the parabola you simply make the x^2 negative to get -x^2 and then to go 2 units down you subtract 2 from the -x^2. This gives you the equation
[tex]g(x)=-x^2-2[/tex]
What is the range of the given function?
{(-2, 0), (-4,-3), (2, -9), (0, 5), (-5, 7)}
O {x|x=-5, -4, -2, 0, 2}
Oy | y=-9, -3, 0, 5, 7}
{x|x = -9, -5, -4, −3,−2, 0, 2, 5, 7}
{y | y=-9, -5, -4, -3, -2, 0, 2, 5, 7}
The range of the function will be (-9, 7). Then the correct option is D.
What are domain and range?The domain means all the possible values of x and the range means all the possible values of y.
The function is given in the form of the pair of points.
{(-2, 0), (-4,-3), (2, -9), (0, 5), (-5, 7)}
Then the range of the function will be (-9, 7).
Then the correct option is D.
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Hello all! Please help me out with this.
I want to find a certain number which I will call Y. It is four times as great as another number which is two less than twenty. What is Y?
Answer:
Y=72
Step-by-step explanation:
First we set up the equation which is (20-2) times 4=Y
Then we solve, 20 minus 2 is risk since it is in the parentheses. 20 minus 2 is 18.
After that 18 times 4 is 72
Y=72
A scientist studying insects start with a population of 10. the population triples every hour. how many insects wil there be after 20 minutes
The question is asking you to understand that a population which is raised by a common ratio over a certain time period follows an exponential pattern. See:
After 1 hour, the population is 3*10
After 2 hours, the population is 3*3*10
After 3 hours, the population is 3*3*3*10
.....
This can be generalised as a function of t, the time in hours:
f(t) = (3^t) * 10
Since the function determines the population at a given time, it is more prudent to replace f(t) with P, the population after time t:
P = (3^t) * 10
Since 20 minutes is equal to (1/3) hours, t can be substituted for (1/3) in order to calculate the population size after 20 minutes:
P = (3^(1/3)) * 10 = 14.4224957031 ≈ 14
Therefore the population after 20 minutes is 14.
Maximiza a: z=5x+2y
Sujeta a: x+2y≤12
x+y≥4
x,y ≥0
Answer:
hi
Step-by-step explanation:
no
x2y, for x = 3 and y = 6
Answer:
12
Step-by-step explanation:
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{ASSUMING}[/tex]
[tex]\mathsf{x^{2y}}[/tex]
[tex]\huge\textbf{IF SO, FOLLOW THESE STEPS TO}\\\huge\textbf{SOLVE FOR YOUR RESULT}[/tex]
[tex]\mathsf{x^{2y}}\\\mathsf{= 3^{2(6)}}\\\mathsf{= 3^{2\times6}}\\\mathsf{= 3^{12}}\\\mathsf{= 3\times3\times3\times3\times3\times3\times3\times3\times3\times3\times3\times3}\\\mathsf{= 9\times9\times9\times9\times9\times9}\\\mathsf{= 81\times81\times81}\\\mathsf{= 6,561\times81}\\\mathsf{= 531,441}[/tex]
[tex]\huge\text{Therefore, your answer should be: \boxed{\mathsf{531,441}}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\huge\boxed{\frak{Amphitrite1040:)}}[/tex]please i need this fast
Answer:
the second option
Step-by-step explanation:
UV and AB are at a right Angie to reach other.
and
UV is parallel to ZY
Find the equation of the line parallel to y = 2x - 4 that runs through the point (-2, 4).
y = 1/2x + 4
y = 2x + 8
y = 2x + 4
y = -1/2x + 8
Answer:
y = 2x + 8
Explanation:
Equation: y = 2x - 4
Comparing it with slope intercept form "y = mx + b" where 'm' is slope and 'b' is y-intercept. This function has slope: 2 and y intercept of -4
Parallel lines has same slope.
Passes through (-2, 4)
[tex]\sf y - y_1 = m(x - x_1)[/tex]
[tex]\sf y - 4 = 2(x - (-2))[/tex]
[tex]\sf y - 4 = 2(x + 2)[/tex]
[tex]\sf y = 2x + 4 + 4[/tex]
[tex]\sf y = 2x + 8[/tex]
Answer: y = 2x + 8
Step-by-step explanation:
This is a fairly simple problem. I hope this helps!
So, to find a line parallel, it's simple, and it only requires that for the other line to have the same slope and a different y-intercept (or same "m" and different "b" in the equation y = mx + b).
To find a line parallel to a point is slightly harder, but don't worry, I'll teach you how.
To do this, we have to recognize the point. The point given is (-2, 4) where -2 is the x-value and 4 is the y-value. Remember, points always go by (x, y).
So, we set y to 4, and we set x to -2.
When testing out lines that are parallel to a point, you must always be careful that you end up with the same value. Such as 4 = 4 or 8 = 8, etcetera.
We get:
4 = 2 × (-2) + 8
Simplifying, we get:
4 = -4 + 8
Adding -4 to 8, we get 4:
4 = 4
Thus, y = 2x + 8 is parallel to the point (-2, 4)
A researcher believes that singing to plants causes them to grow an extra 0.5 inches on average per month. She tests this with 20 of her own plants by bringing 10 to work and leaving 10 at home. The ones at home she sings to each day, and she does not sing to the ones at her work.
The control variable that can be used in the research experiment is to ensure that the two set of plants will receive the same amount of water and light.
How to depict the control variable?According to the information, the researcher wants to seek whether singing to plants causes them to grow taller.
The explanatory variable is singing. Therefore, the researcher can control some of the variables to determine the explanatory variable on the growth of the plant.
Hence, the control variable that can be used in the research experiment is to ensure that the two set of plants will receive the same amount of water and light.
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Need some help ASAP and work shown. Ps; If you answer I’ll mark you brainliest.
Answer:
h ≈ 37 ft
Step-by-step explanation:
let x be the height of the tree from the viewers eye level
using the tangent ratio, then
tan24° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{72}[/tex] ( multiply both sides by 72 )
72 × tan24° = x
then
height of tree = x + 5 = 72tan24° + 5 ≈ 37 ft ( to the nearest foot )
Answer:
37ft
Step-by-step explanation:
It can be considered as a right angle triangle. The height of the tree is equal to the product of tan(24°) and the distance between the man and tree+5 Therefore
⇒h=tan(24°)*72+5⇒h≈32ft+5⇒h≈37In conclusion
The height of the tree is 37ft
Factor the following expression. x² - 10x +9 2 Factor the following expression . x² - 10x +9 2
Answer:
x² - 10x + 9 = (x-9)(x-1)
the factored expression would be (x-9)(x-1)
A community center has an L-shaped swimming pool that is 10 feet deep. What is the volume of the pool?
The total volume of the pool is 1040 cubic ft. The correct answer is option B.
The complete question is given below:-
The community centre has an L-shaped swimming pool that is 10 feet deep. What is the volume of the pool
1200
1040
640
880
What is volume?Volume is defined as the space covered by any solid body in the three-dimensional plane.
We know that the total volume of the pool is the sum of the volume of two rectangular prisms.
One is a rectangular prism with dimensions 20ft.×10ft.×4ft.
and the other rectangular prism has dimensions: 6ft.×4ft.×10ft.
Hence, the volume of a first rectangular prism is:
20×10×4=800 cubic ft.
The volume of a second rectangular prism is:
6×4×10=240 cubic ft.
Hence, the total volume of the pool is:
800+240=1040 cubic ft.
Therefore the total volume of the pool is 1040 cubic ft. The correct answer is option B.
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A pyramid has a square base with sides of length s. The height of the pyramid is equal to 1/2 of the length of a side on the base. Which formula
represents the volume of the pyramid?
A. V = 1/12 s ^3
B. V=1/6 s ^3
C. V=1/3s ^3
D. V=3s³
E. V=6s³
Answer:
V = (s^3)/6 (Answer B )
Step-by-step explanation:
If the length of a side of the base is s, then the height of the pyramid is half that, or s/2.
The volume of a pyramid with a square base and base side length S and height H is V = (1/3)(S^2)(H).
In this case, the formula is V = (1/3)(s^2)(s/2), or V = (s^3)/6 (Answer B)
I walked 280m of the road from the park to the station. This is equivalent to 9/10 of the total distance. How many meters is the total distance from the park to the station?
Answer:
311 1/9 or 311.11 mStep-by-step explanation:
Let the total distance be x.
We have:
9/10 of x is 280 m.Set this as equation and solve for x:
(9/10)x = 280x = 280 ÷ 9/10x = 280 × 10/9x = 2800/9 = 311 1/9 or 311.11 mTotal distance from the park to the station is approximately 311 m.
Let that be d
9/10 of d=2809/10d=280d=280(10/9)d=311.1mPLEASE HELP
True or false
False
Step-by-step explanation:
Answer:
true.
Step-by-step explanation:
[tex]b^t=\frac{n}{a}; \ = > \ t=log_b\frac{n}{a}; \ = > \ t=\frac{log\frac{n}{a}}{logb}.[/tex]
A 400 second song is divided into three parts the beginning the middle and the end the beginning is one sixth as long as the end
The beginning, middle and end part of the song is 40, 120 and 240 seconds respectively
How to find equation of the unknownlet
x = end partBeginning = 1/6xMiddle = 1/2xTotal = 400 secondsx + 1/6x + 1/2x = 400
(6x+x+3x) / 6 = 400
10/6x = 400
x = 400 ÷ 10/6
x = 400 × 6/ 10
= 2400 / 10
= 240 seconds
Therefore,
End part = x
= 240 seconds
Beginning = 1/6x
= 1/6 × 240
= 40 seconds
Middle = 1/2x
= 1/2 × 240
= 120 seconds
Complete question:
The beginning is one sixth as long as the end. The middle is one half as long as the end. Find the length of each part
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8. Mrs. Reyes has 4 mangoes, 5 guavas, and 2 bananas in her basket. What is the probability that one can pick a guava?
A. 2 out of 11 or 2/11
B. 5 out of 11 or 5/11
C. 4 out of 11 or 4/11
9. What is the probability of choosing a blue paper if there are 3 red papers and 4 blue papers in the box?
A. 4 out of 7 or 4/7
B. 3 out of 7 or 3/7
C. 0 out of 7 or zero probability
10. In the numbers 2,2,2,5,5,2,3 What is the probability of getting 2?
A. 2/7
B. 3/7
C. 4/7
8) 5 + 4 + 2 = 11
Since there are 5 guavas, the correct answer is B. 5/11
9) 3 + 4 = 7
Since there are 4 blue papers, the correct answer is A. 4/7
10) Count all of the numbers and add them. There are 4 twos, 2 fives, and 1 three:
4 + 2 + 1 = 7
Since there are 4 twos, the correct answer is C. 4/7
A culture started with 1,000 bacteria. After 4
hours, it grew to 1,100 bacteria. Predict how
many bacteria will be present after 12 hours.
Round your answer to the nearest whole
number.
Answer:
1331 bacteria
Step-by-step explanation:
Number bact = 1000 e^kt
1100 = 1000 e^k4
shows k = .023827545
Now for 12 hours
x = 1000 e^(.023827545 * 12)
= 1331
i realized my answer is wrong.
Drag the tiles to the correct boxes to complete the pairs. Match each function to its domain and range.f(x)=5x-3
f(x) is a linear function, so both the domain and the range are the set of real numbers.
2. A line goes through the points (4, 8) and (-4, 6).
(a) What is the slope of the line? Show your work
(b) Write the equation of the line in point-slope form. Show your work
(c) Write the equation of the line in slope-intercept form. Show your work.
Any body can help?
(a) [tex]\frac{6-8}{-4-4}=\frac{-2}{-8}=\boxed{1/4}[/tex]
(b) Using the point (4,8), we get [tex]\boxed{y-8=\frac{1}{4}(x-4)}[/tex]
(c) Rearranging into slope-intercept form,
[tex]y-8=\frac{1}{4}(x-4)\\\\y-8=\frac{1}{4}x-1\\\\\boxed{y=\frac{1}{4}x+7}[/tex]
