The given sum can be expressed as:
f(x) = sin(x)/3 + sin(2x)/5 + sin(3x)/7 + sin(4x)/9 + ...
We can simplify this expression using the identity:
sin(nx) = Im(e^(inx))
where Im(z) denotes the imaginary part of complex number z, and e^(ix) is the complex exponential function.
Using this identity, we can rewrite f(x) as:
f(x) = Im [e^(ix)/3 + e^(2ix)/5 + e^(3ix)/7 + e^(4ix)/9 + ...]
We can then use the formula for the sum of an infinite geometric series:
1/(1 - r) = 1 + r + r^2 + r^3 + ...
where |r| < 1.
In our case, we have:
r = e^(ix)/3
So the sum can be written as:
f(x) = Im [1/(1 - e^(ix)/3)]
To evaluate this expression, we can use the complex conjugate:
1/(1 - e^(ix)/3) = (1 - e^(-ix)/3) / (1 - 2cos(x)/3 + 1/9)
We can then use the identity:
Im(z) = (z - z*) / (2i)
where z* is the complex conjugate of z.
Using this identity, we can simplify f(x) to:
f(x) = (1/2i) [(1 - e^(-ix)/3) / (1 - 2cos(x)/3 + 1/9) - (1 - e^(ix)/3) / (1 - 2cos(x)/3 + 1/9)*]
This simplifies to:
f(x) = (3/4) [sin(x)/(1 - 2cos(x)/3 + 1/9) - sin(-x)/(1 - 2cos(x)/3 + 1/9)*]
Since sin(-x) = -sin(x), we have:
f(x) = (3/2) [sin(x)/(1 - 2cos(x)/3 + 1/9)]
This is the closed form of the sum f(x).
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helpp how do i solve this
The correct solution is shows in below.
What is mean by Fraction?A fraction is a part of whole number, and a way to split up a number into equal parts. Or, A number which is expressed as a quotient is called fraction. It can be written as the form of p : q, which is equivalent to p / q.
Given that;
1) 3/4 - 1/5 = 11/20
2) 3 1/3 - 4/7 = 2 16/21
3) 3 3/7 - 2 2/3 = 16/21
Now,
Simplify the fraction as;
1) 3/4 - 1/5 = 11/20
LHS;
⇒ 3/4 - 1/5
⇒ (15 - 4) / 20
⇒ 11/20
⇒ RHS
2) 3 1/3 - 4/7 = 2 16/21
⇒ 10/3 - 4/7
⇒ (70 - 12) / 21
⇒ 58 / 21
⇒ 2 16/21
⇒ RHS
3) 3 3/7 - 2 2/3 = 16/21
LHS;
⇒ 3 3/7 - 2 2/3
⇒ 24/7 - 8/3
⇒ (72 - 56)/21
⇒ 16/21
⇒ RHS
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hey free 100 points :) (not really) please help!
Complete the square in the quadratic function f given by
f(x) = 2x2 - 6x + 4
Answer: x = 2, 1
Step-by-step explanation:2x^2 - 6x + 4 = 0
Firstly, simplify. Divide by 2 to get
2(x^2-3x+2) = 0
Move the constant to the other side.
2(x^2-3x) = -2
Divide the x term by 2 and square it
-3/2^2 = 9/4
Add this result to both sides.
2(x^2-3x+9/4) = 1/4.
Put the perfect sq on both sides
2(x-3/2)^2=1/4
Then solve the equation:
take the sqrt of both sides and isolate x for each solution
x-3/2= +/-1/2
x = 3/2 + 1/2
x= 3/2-1/2
x= 2, 1
A phone company offers a monthly data plan for $10 a month. The planincludes 2 megabytes of data, and charges $0.10 per megabytes above the2 megabytes of data. Write a piecewise-defined function forM(x), the cost forxmegabytes of data used in a month.
The piecewise-defined function for M(x), the cost for x megabytes of data used in a month [tex]$m(x)=\left\{\begin{array}{c}10, i f x \leq 2 \\ 0.1 x+9.8, i f x > 2\end{array}\right.$[/tex].
What is a function?A function in mathematics from a set X to a set Y allocates precisely one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Data exceeding 2 MB is charged at a rate of $0.10 per MB; one-time fee is $10;
The cost per month for x MB of data is:
Up to 2 Mb of data utilized; $10
Data use of over 2 MB
One off charge = $ 10
Charge for data over 2 Mb is $ 0.10 per MB
The monthly charge for x MB data is:
Data ta used up to 2 Mb
$ 10
Data used more than 2 Mb
m(x) = 10 + 0.1(x-2)
m(x) = 10 + 0.1 x-0.2
m(x) = 0.1 x + 9.8
So the function is:
[tex]$m(x)=\left\{\begin{array}{c}10, i f x \leq 2 \\ 0.1 x+9.8, i f x > 2\end{array}\right.$[/tex]
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WILL GIVE BRAINLIEST
Line segment MN joins the midpoints of two sides of the triangle below. The length of MN is half the length of the base of the triangle. What is the value of x in centimetres?
Answer:
48 centimeters
Step-by-step explanation:
This deals with the midline theorem.
The midline is defined by a seg that is joined by the midpoints of 2 sides.
The midline is half of the base, so if it is 10, the base is 20 cm.
You see that 26 cm is 1/2 of the hypotenuse because it is bisected, sot he whole thing is 26(2) or 52.
Now you have to solve for x using the Pythagorean theorem.
20^2 (400) + x^2 = 52^2 (2704)
Subtracted 400 from both sides to get x^2 by itself.
x^2 = 2304
Take the square root of both sides to isolate x (take the principle root)
x= 48 centimeters.
Hope this helps!
Steve’s savings account has a balance of $1287. after 4 years, what will the amount of interest be at 6% compounded semiannually? a. $161.53 c. $308.88 b. $337.81 d. $343.33
Answer:
d. $343.33
Step-by-step explanation:
We use the equation
A = P(1 + r)^t
P = $1287
r = 6% = 0.06
t = 4 years
Let's solve
A = 1287( 1 + 0.06)^4
A = 1287(1.06)^4
A = $1630.33
$1630.33 - $1287 = $343.33
So, the answer is D
6. Which of the following is not a way to name the plane?
b. Plane IGF
a. Plane P
c. Plane HDG
d. Plane HDF
Answer: P
Step-by-step explanation:
Is 7/8 greater than 2/5?
Answer:
Step-by-step explanation:
Yes 7/8= 0.875. 2/5 is equal to 0.40
Elena and Jada each make money by helping out their neighbor. Elena babyit. Her earning are given by the equation y=8. 40x, where x repreent how many hour he work and y repreent how much money he earn. Jada earn $7 per hour mowing her neighbor' lawn. How long would it take to make each peron 150$?
If Elena earn y = 8.40x dollars where x represent how many hours she work and Jada earn $7 per hour, it would take Elena approximately 17.86 hours and Jada approximately 21.43 hours to earn $150
To find out how many hours it would take for Elena to earn $150, we need to solve the equation:
y = 8.4x
150 = 8.4x
x = 150/8.4
x ≈ 17.86 hours
So it would take Elena approximately 17.86 hours to earn $150.
To find out how many hours it would take for Jada to earn $150, we divide 150 by her hourly rate:
= $150 /( $7/ hour)
≈ 21.43
So it would take Jada approximately 21.43 hours to earn $150.
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What is the name of a triangle when all 3 sides are the same length?
The name of a triangle when all 3 sides are the same length is Equilateral Triangle
What is an Equilateral Triangle ?A triangle is said to be equilateral if each of its three sides is the same length. In the well-known Euclidean geometry, an equilateral triangle is also equiangular, meaning that each of its three internal angles is 60 degrees and congruent with the others.
Equilateral Triangle Properties
Equal on all three sides.
The sum of the three congruent angles is 60 degrees.
It has three regular sides and is a polygon.
The equilateral triangle is divided into equal halves by the perpendicular that is traced from its vertex to the other side. Additionally, the perpendicular's origin vertex's angle is split into two equal angles of 30 degrees each.
The centroid and ortho-centre are located at the same location.
The median, angle bisector, and altitude of an equilateral triangle are all the same.
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Can all quadratic functions be written in factored form?
No, not all quadratic equations can be solved by factoring. This is because not all quadratic expressions, ax2 + bx + c, are factorable.
What are quadratic functions?A quadratic function has the formula f(x) = ax2 + bx + c, with a, b, and c being integers and a not equal to zero. A parabola is a curve that represents the graph of a quadratic function. Parabolas can have openings that are upward or downward, different "widths" or "steepnesses," but they all share the same fundamental "U" shape. In mathematics, a quadratic issue is any problem that involves multiplying a variable by itself, often known as "squaring." This terminology comes from the fact that a square's area is equal to its side length multiplied by itself. The Latin term quadratum, which means square, is where the word "quadratic" derives. No, factoring cannot be used to solve every quadratic problem. Because not all quadratic expressions, ax2 + bx + c, are factorable, this is the case.To learn more about quadratic functions refer to:
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For which equation is b = 5 not the solution?
8- b = 13
B + 6 = 11
B + 4 = 9
12 - b =7
Answer:8-b=13
Step-by-step explanation:
8-b=13
-b=13-8
-b=5
b=-5
What is the domain and range for this function?
Which transformations are nonrigid transformations?
a. rotation
b. dilation
c. reflection
d. translation
e. stretch
Dilation and Stretch are non-rigid transformations.
Non - Rigid transformation:
Non-rigid transformations can change the size or shape of the archetype, or both size and shape. The two transformations Strain and Shear are not rigid. The resulting image of the transform is changed in size, shape, or both.
A non-rigid transformation is a transformation that changes the length or angle of a shape's sides. Nonrigid transformations are dilation and stretching.
All types of transformations except dilation are rigid transformations.
Dilation transformations are divided into two types
StretchingCompressionTherefore non-rigid transformations are Dilation and Stretching
Dilation:
Dilation (usually denoted by ⊕) is one of the fundamental operations in mathematical morphology. Originally developed for binary images, it was expanded first to grayscale images and then to full meshes. Stretch operations typically use a structuring element to examine and expand the shapes contained in the input image.
Strechting:
A non-rigid transformation changes the size or shape of an object. Resizing (stretching horizontally, vertically, or both) is a non-rigid transformation.
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How do you solve inequalities in 8th grade?
1. Add the same number to both sides.
2. Subtract the same number from both sides.
3. Multiply both sides by the same positive number.
4. Divide both sides by the same positive number.
5. Multiply both sides by the same negative number and reverse the sign.
The sum of two numbers is 56. The difference between the two numbers is 12. What are the two numbers?
Answer:
Step-by-step explanation:
x+y=56
x+12=y
2x+12=56
2x=44
x=22
y=56-22
y=34
x=22
hope this helps
Answer:
34 and 22
Step-by-step explanation:
x + y = 56 Add the two equations together
x - y = 12
2x = 56 Divide both sides by 2
x = 34
substitute 34 for x in either of the two original equations to solve for y
x + y = 56
34 + y = 56 Subtract 34 from both sides
y = 22
Check:
x + y = 56
34 + 22 = 56 Checks
x - y = 12
34 - 22 = 12
12 = 12 Checks
Rosie enters the 120-step contest. She takes 6 jumps and overshoots the 120-step mark by 6 steps. How long was each of Rosie’s jumps?
The length of each of Rosie's jumps, given that she overshoots the 120 - step mark by 6 steps, is 21 steps.
How to find the length of the jumps?Rosie is involved in the 120 - step contest and when she took 6 jumps, she overshot the 120 - step mark by 6 steps. This simply meant that she took 6 more steps over the 120 mark, thanks to her jump length.
The number of steps she took in total is therefore :
= 120 + 6
= 126 steps
This then means that the length of each jump was :
= Number of steps in total / Number of jumps
= 126 / 6
= 21 steps
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A ceiling fan has 19-inch blades(so the radius of the circular fan is 19 inches). Suppose the dan turns at a rate of 69 revolutions per minute.
a) Find angular speed of the fan in radians per minute
b) Find the linear speed of the tip of a blade in inches per minute
Angular speed of the fan in radians per minute is 421 (crad /min) , The linear speed of the tip of a blade in inches per minute is
What the linear speed and angular speed?Suppose the dan turns at a rate of 69 revolutions per minute.
Angular speed of the fan in radians per minute is
According to the problem
w = 2zn
(a) w = 2z * 67
= 134z
≈ 421 crad /min)
Angular speed of the fan in radians per minute is 421 (crad /min)
The linear speed of the tip of a blade in inches per minute is
(b) v = 2z * 19 * 67
{ v = 2zrn}
=2546z
7988(cin / m * in)
The linear speed of the tip of a blade in inches per minute is 7988(cin / m * in)
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What are the 7 types of angles Class 7?
Answer:
Step-by-step explanation:
Zero Angle (0° in Measure)
Acute Angle (0 to 90° in Measure)
Right Angle (90° in Measure)
Obtuse Angle (90 to 180° in Measure)
Straight Angle (180° in Measure)
Reflex Angle (180 to 360° in Measure)
Complete Angle (360° in Measure)
Line m passes through point (5, 8) and has a slope 3/5. What is the equation for the line
The equation for the line is y = 3/5x + 5.
What is equation ?
The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.
A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used.
y = mx + b
8 = 3/5(5) + b
8 = 15/5 + b
b = 5
∴ equation is y = 3/5x + 5
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A baker sells bread for $3 a loaf and rolls for $1 each. The baker needs to sell $27 worth of baked goods by
the end of the day.
Graph the linear equation 3b + r = 27, where b is the number of loaves of bread sold and r is the
number of rolls sold.
Answer:
4 loaves of bread
Step-by-step explanation:
Cost of 1 loaf of bread = $3
Cost of 1 roll = $1
The baker needs to sell both items and it should be worth $24.
To find the linear equation,
Suppose, the baker sells
x
loaves of bread and
y
rolls.
Then the total cost of the sales has to be $24.
So, the linear equation will be:
3
x
+
1
y
=
24
.......................(A)
Now, it is given that the baker sold 12 rolls and we need to know how many loaves he has to sell.
For that, replace the
y
in equation (A) with 12 as
y
represents number of rolls sold.
Then the new equation becomes:
3
x
+
1
×
12
=
24
0r
3
x
=
24
−
12
or
3
x
=
12
Solving for
x
,
x
=
12
3
=
4
loaves
What additional information must be known to prove the triangles similar by SSS?
A)
The lengths of and
B)
∠F ≅ ∠K
C)
No additional information is needed.
D)
∠D ≅ ∠J
The angle also should be known to prove congruency.
What is congruence?It states that two triangles are said to be congruent if they are exact copies of one another and completely cover one another when superposed. In other words, two triangles are congruent if their corresponding sides and angles are equal to those of the first triangle.
A congruence theorem, is AAA?It is ineffective to only know angle-angle-angle (AAA), as this can result in triangles that are similar but not congruent. There are four short cuts that can be used to check whether two triangles are congruent.
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What is the distance, in units, between the y-intercept of f(x) = 1/2x - 6 and the y-intercept of the linear function that passes through the points shown in the table below?
Only in the absence of acceleration, S, is it precisely proportional to time. When there is no outside force acting on the item, its acceleration 'a decreases to zero, and the equation of motion changes to s= ut.
How do you find the distance between y-intercept and x-intercept?Subtract their x-coordinates to get the distance between them. The x-coordinate is the same for every point along a vertical line. Subtract their y-coordinates to calculate the distance between them. Below is a picture of the equation's graph, 2x+3y=6. In this location, the graph departs from the x-axis (3,0). The x-intercept is this particular location.
The point where the graph also deviates from the y-axis (0, 2). In the case of a speed vs time graph, where time is on the x-axis and speed is on the y-axis, the intercept's SI unit will be speed (m/s). x-intercept: You can find x by setting y=0 in the equation y=2x-6. 3.0 is the x-intercept. y-intercept: Enter x=0 in y = 2x - 6 and find x using the formula. It is -6 for the y-intercept.
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In Mr. Ibrahim's classroom, 20 out of 25 students saw a new action movie. What is the ratio of students who saw the movie to students who did not see the movie? (Think, how many students saw the movie and how many students did not see the movie.)
Answer:
20:5
Step-by-step explanation:
There were 25 kids and 20 of them watched the movie, meaning only 5 people did not watch so it is 20:6.
Write the equation of a sine function that has the following characteristics.
1
Amplitude: 4 Period: 7pi Phase shift:
1/5
Answer:
[tex]y=4\sin(\frac{2}{7}x-\frac{2}{35})[/tex]
Step-by-step explanation:
Period: [tex]7\pi=\frac{2\pi}{b}\rightarrow b=\frac{2}{7}[/tex]
Phase Shift:
[tex]\displaystyle \frac{c}{b}=\frac{1}{5}\\\\\rightarrow\frac{c}{\frac{2}{7}}=\frac{1}{5}\\\\\rightarrow \frac{2}{7}=5c\\ \\\rightarrow \frac{2}{35}=c[/tex]
Final equation:
[tex]y=4\sin(\frac{2}{7}x-\frac{2}{35})[/tex]
h Find the lateral surface area of the net. 7 1.5 1.5 2 ft 2pt 5 1.5 ft r 5 ft OManeuvering the Middle LLC, 2016
On solving the provided question we can say that the Lateral Surface area=396cm2
What is Lateral Surface Area?The surface area of a three-dimensional object less the area of its bases is the item's lateral surface area. All of an object's sides, excluding its base and top, are considered its lateral surface. The size of the lateral surface is referred to as its area. This must be distinguished from the total surface area, which consists of the base and top areas as well as the lateral surface area.
The Lateral surface area of cylinder
=2πrh
2π×7×18=792cm2
When the height is reduced to half then using the same formula for the lateral surface area
we have
2π×7×9=396cm2
Lateral Surface area=396cm2
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On solving the provided question we can say that the Lateral Surface area = 396 cm²
What is Lateral Surface Area?The surface area of a three-dimensional object less the area of its bases is the item's lateral surface area. All of an object's sides, excluding its base and top, are considered its lateral surface. The size of the lateral surface is referred to as its area. This must be distinguished from the total surface area, which consists of the base and top areas as well as the lateral surface area.
The Lateral surface area of cylinder
=2πrh
2π×7×18=792cm2
When the height is reduced to half then using the same formula for the lateral surface area
we have
2π×7×9=396cm2
Lateral Surface area=396cm2
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The work shows the first steps of writing a partial fraction decomposition. Startfraction 3 x + 9 over (x + 1) (x minus 5) endfraction = startfraction a over x + 1 endfraction + startfraction b over x minus 5 endfraction 3 x + 9 = a (x minus 5) + b (x + 1) 3 x + 9 = a x minus 5 a + b x + b what is the partial fraction decomposition in terms of x?.
The partial fraction decomposition in terms of x for the given equation is:
3x + 9 = (a(x-5) + b(x+1))
To solve for a and b, we need to multiply both sides by (x+1)(x-5). This will give us:
3x² + 4x - 45 = ax - 5a + ax + b
Now we can move all the terms with an x to one side and all the terms without an x to the other side. This will give us:
3x² + 4x - ax + b = 45 - 5a - b
We can rearrange this equation to get the following:
3x² - ax + (4 + b) = (45 - 5a - b)
Now we can use the quadratic formula to solve for a and b:
a = (4 + b) + √(4²- 4×3×(45 - 5a - b))/ (2×3)
b = (45 - 5a - (3x² - ax)) / (4 + b)
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Answer:
It's C
Step-by-step explanation:
StartFraction 3 x + 9 Over (x + 1) (x minus 5) EndFraction = StartFraction negative 1 Over x + 1 EndFraction + StartFraction 4 Over x minus 5 EndFraction
Use a trigonometric ratio to solve for c. Round to two decimal places as necessary!!
HELP, and can someone explain how to get the answer?
Given that the one side of a right-angled triangle is 12 and its angle is 19, the value of c is 36.85.
what is trigonometry ?Trigonometry, a branch of mathematics, studies how triangle side lengths and angles relate to one another. Around the third century BC, the use of geometry in astronomical study led to the first appearance of the area. The study of specific trigonometric functions and how they could be applied to calculations is a topic covered in the branch of mathematics known as precise techniques. Trigonometry includes six well-known trigonometric operations. The names and acronyms for each are sine, cosine, tangent, cotangent, secant, and cosecant (csc). Trigonometry is the study of triangle properties, specifically of right triangle properties. But geometry is the study of geometric figures and their features.
given
a/sinA = c / sinB
12 / sin19 = c / sin90
36.85 = c /1
c = 36.85
Given that the one side of a right-angled triangle is 12 and its angle is 19, the value of c is 36.85.
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1. Find the midpoint of the line segment joining (–4, –2) and (2, 8).
The midpoint of the line segment joining (–4, –2) and (2, 8) is calculated to give: (-1, 3)
What is a line's midpoint?The centroid of a line segment is its middle point. Since it is equally far from both ends, it divides the region in half.
How to determine a line segment's midpointIn coordinate geometry, the midpoint formula is the method for calculating the center point of a straight line given its endpoints' coordinates.
The formula is of the form
X= (x₂ + x₁)/2
Y = (y₂ + y₁)/2
For points (–4, –2) and (2, 8).
X= (-4 + 2)/2 = -1
Y = (-2 + 8)/2 = 3
The midpoint is solved to be (-1, 3)
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if you add Natalie's age and Fred's age, the result is 38. If you add Fred's age to 3 times Natalie's age, the result is 66. Write and solve a system of equations to find how old Fred and Natalie are.
a) The system of equations to find Fred and Natalie's ages are:
x + y = 383x + y = 66.b) Solving the system of equations shows that their respective ages are:
Fred = 24Natalie = 14.What is a system of equations?A system of equations is two or more equations simultaneously or concurrently solved.
We call this a system of equations or simultaneous equations because they are solved at the same time.
Let Natalie's age = x
Let Fred's age = y
The total ages of Natalie and Fred = 38
Equation 1:x + y = 38
Let Natalie's age = 3x
Let Fred's age = y
The total ages of Natalie and Fred = 66
Equation 2:3x + y = 66
Subtract Equation 1 from Equation 2:
3x + y = 66
-
x + y = 38
2x = 28
x = 14
y = 38 - 14
y = 24
Thus, Fred is 24 years old while Natalie is 14 years.
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The table below represents the number of power boats registered in the
given year. Using the equation for the line of best fit, which is a
good prediction for the number of powerboat registrations in 2013?
Therefore , the solution of the given problem of equation comes out to be 1225 boats registrations done in year 2013.
What is equation?The equal letter (=) is used to signify equivalence between the statements in a mathematical formula. Using mathematical equation, which are declarations of reality, it is shown that many mathematical variables are equivalent. For instance, the equal sign in the equation y + 6 = 12 divides the values 12 or b + 6 into two halves. The number of words that each side of a symbol corresponds to can be measured. Typically, a symbol's meaning is at odds with itself.
Here,
Given :
Let x be the number of boats:
t is number of years
In 1996 and 1997
=> 797 -751 = 46 boats increased
In 1997 and 1998
=> 806 -797 = 9 boats increased
So ,
the equation which can signify the given information :
=> 2002 + t = 923 +x
So ,At 2013
=> 2013 = 1225 boats registrations done.
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