Recall that for [tex]|x|<1[/tex], we have the convergent geometric series
[tex]\displaystyle \sum_{n=0}^\infty x^n = \frac1{1-x}[/tex]
Now, for [tex]\left|\frac x5\right| < 1[/tex], we have
[tex]\dfrac1{5 - x} = \dfrac15 \cdot \dfrac1{1 - \frac x5} = \dfrac15 \displaystyle \sum_{n=0}^\infty \left(\frac x5\right)^n = \sum_{n=0}^\infty \frac{x^n}{5^{n+1}}[/tex]
Integrating both sides gives
[tex]\displaystyle \int \frac{dx}{5-x} = C + \int \sum_{n=0}^\infty \frac{x^n}{5^{n+1}} \, dx[/tex]
[tex]\displaystyle -\ln(5-x) = C + \sum_{n=0}^\infty \frac{x^{n+1}}{5^{n+1}(n+1)}[/tex]
If we let [tex]x=0[/tex], the sum on the right side drops out and we're left with [tex]C=-\ln(5)[/tex].
It follows that
[tex]\displaystyle \ln(5-x) = \ln(5) - \sum_{n=0}^\infty \frac{x^{n+1}}{5^{n+1}(n+1)}[/tex]
or
[tex]\displaystyle \ln(5-x) = \boxed{\ln(5) - \sum_{n=1}^\infty \frac{x^n}{5^n n}}[/tex]
12. Diameter,
A = 72Pie
(Circle)
Answer:
6.98
Step-by-step explanation:
I think that you are looking for the diameter and you are given the area of a circle.
a =[tex]\pi r^{2}[/tex] You are given pi, so [tex]r^{2}[/tex] = 72.
That means that r = [tex]\sqrt{72}[/tex] or a rounded answer of 8.49 rounded to the hundreds place. The diameter is equal to 2 radius, so the diameter rounded is 16.98
Which linear inequality is represented by the graph
Answer:
the second choice
Step-by-step explanation:
The last two choices could not be correct, because it would need a dashed line. The line under the greater than or less than symbol mean that the line is included in the answer and it will be a solid line like you see in the picture. For both of the top two answers the slope that they y intercept is correct. You just have to decide is the symbol will be greater than or equal to or less than and equal to.
If we put 0 in x for in the form y = mx + b, what would y be?
y = 1/3 x - 1
y = 1/3(0) - 1
y = -1
The ordered pair is (0,1). That is the part that is shaded so that shows us that the second answer is correct.
Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.
ABIs parallel to CD, and EFis perpendicular to AB.
The number of 90° angles formed by the intersections of EFand the two parallel lines ABand CDis
Answer:
8
Step-by-step explanation:
The four angles formed at each intersection of lines are all 90°.
The total number of 90° angles at the two intersections is 2×4 = 8 90° angles.
The types of transformations of geometric figures in the coordinate plane can be described as a slide, a flip, or a turn. What are the other names used to identify these transformations?
The other names used to identify the transformation of geometric figures is translation or reflection and rotation.
Given that the types of transformations of geometric figures in the coordinate plane can be described as a slide, a flip, or turn.
We are required to find the other names used to identify the above transformations.
The other names used to identify the transformation of geometric figures is translation or reflection and rotation.
Translation means the displacement of a figure from one place to another. In translation a figure can move upward, downward,right , Iift or anywhere in the coordinate system. In translation, the position of the object changes, its size remains the same.
Reflection is a type of transformation that flips a shape in a mirror line so that each point is the same distance from the mirror line as its reflected point.
Rotation is a motion of a space that preserves at least one point.
Hence the other names used to identify the transformation of geometric figures is translation or reflection and rotation.
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A coordinate grid with 2 lines. The first line passes through (0, negative 5) and (negative 5, 0). The second line passes through (0, negative 5) and (negative 2, 1).
What is the solution to the system of equations?
Answer:
(0, -5)
Step-by-step explanation:
The two different lines have point (0, -5) in common, so the solution is (0, -5).
10 by 28 + - 32 by 49 rational number
pls answer it
thank you
By simplifying the fraction 10/28 ÷ 32/49 exists 35/64.
How to simplify fractions?The first stage of dividing fractions exists to estimate the reciprocal (reverse the numerator and denominator) of the second fraction. Subsequently, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if required.
To find the reciprocal of the divisor
Reciprocal of 32/49 = 49/32
Now, multiply it by the dividend, then we get
So, 1028 ÷ 3249 = 10/28 × 49/32
= (10 × 49)/(28 × 32) = 490/896
After decreasing the fraction, the answer exists at 35/64.
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divide $800 between kofi and kweku so that kofi gets three times what kweku gets
Answer:
600 and 200
Step-by-step explanation:
kofi : kweku is 3 :1
so Kofi gets 3 out of ( 3 +1) = 3/4 of 800 = 3/4 * 800 = 600
kweku get s the rest 800- 600 = 200
Use the given transformation x=4u, y=3v to evaluate the integral. ∬r4x2 da, where r is the region bounded by the ellipse x216 y29=1
The Jacobian for this transformation is
[tex]J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 3 \end{bmatrix}[/tex]
with determinant [tex]|J| = 12[/tex], hence the area element becomes
[tex]dA = dx\,dy = 12 \, du\,dv[/tex]
Then the integral becomes
[tex]\displaystyle \iint_{R'} 4x^2 \, dA = 768 \iint_R u^2 \, du \, dv[/tex]
where [tex]R'[/tex] is the unit circle,
[tex]\dfrac{x^2}{16} + \dfrac{y^2}9 = \dfrac{(4u^2)}{16} + \dfrac{(3v)^2}9 = u^2 + v^2 = 1[/tex]
so that
[tex]\displaystyle 768 \iint_R u^2 \, du \, dv = 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2 \, du \, dv[/tex]
Now you could evaluate the integral as-is, but it's really much easier to do if we convert to polar coordinates.
[tex]\begin{cases} u = r\cos(\theta) \\ v = r\sin(\theta) \\ u^2+v^2 = r^2\\ du\,dv = r\,dr\,d\theta\end{cases}[/tex]
Then
[tex]\displaystyle 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2\,du\,dv = 768 \int_0^{2\pi} \int_0^1 (r\cos(\theta))^2 r\,dr\,d\theta \\\\ ~~~~~~~~~~~~ = 768 \left(\int_0^{2\pi} \cos^2(\theta)\,d\theta\right) \left(\int_0^1 r^3\,dr\right) = \boxed{192\pi}[/tex]
Prove that the number A= [tex]20^{8^{2014} } }+113 is composite
Cheese proof:
We can just prove that it is divisible by 3, which means that it is composite. We can use modular exponentiation, where if [tex]a \equiv b \pmod{n}[/tex] then [tex]a^x \equiv b^x \pmod{n}[/tex]. In this case, [tex]{-1}^{8^{2014}} \equiv 20^{8^{2014}} \pmod{3}[/tex]. This is much easier to calculate! Since [tex]8^{2014}[/tex] is even, [tex]-1^{8^{2014}}=1[/tex], meaning that now we only need to prove that [tex]0\equiv(1+113) \pmod{3}[/tex], which is obviously true.
Pythagorean Theorem
In a right triangle, the square of the _____ equals the sun of the squares of the ______.
Answer: In a right triangle, the square of the Hypotenuse equals the sun of the squares of the other two sides.
Step-by-step explanation: a^2+b^2=c^2
Answer: the Pythagoras theorem states that in a right angled triangle, the square of THE HYPOTENUSE is equal to the sum of the squares of the OTHER TWO SIDES.
Step-by-step explanation: :)
What is the fiftieth term in the sequence 5, 7, 9, 11, 13 …?
The 15th term in the given A.P. sequence is a₁₅ = 33.
According to the statement
we have given that the A.P. Series with the a = 5 and the d is 2.
And we have to find the 15th term of the sequence.
So, for this purpose we know that the
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
And the formula is a
an = a + (n-1)d
After substitute the values in it the equation become
an = 5 + (15-1)2
a₁₅ = 5 + 28
Now the 15th term is a₁₅ = 33.
So, The 15th term in the given A.P. sequence is a₁₅ = 33.
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6x+4y-3z,7x-11y-9z,14x+8y-6z
Step-by-step explanation:
i donot know
sorry about this
The cross-sectional areas of a triangular prism and a right cylinder are congruent. the triangular prism has a height of 5 units, and the right cylinder has a height of 5 units. which conclusion can be made from the given information?
The conclusion can be made from the given information
The volume of the triangular prism is equal to the volume of the cylinder
Given that there are two figures
1. A right triangular prism
2. Right cylinder
The area of the cross-section of the prism is equal to the Area of a cross-section of the cylinder.
Let this value be A.
Also given that the Height of prism = Height of cylinder = 6
The volume of a prism is will be :
[tex]V _{prism} = cross section area \times height[/tex]
[tex]V _{prism} = A \times 6 = 6A[/tex] (1)
The Cross section of the cylinder is a circle.
hence the Area of the circle will be:
Area of cross-section, A = [tex]\pi \times r^2[/tex]
so, the Volume of the cylinder will be :
[tex]V _{cylinder} = \pi \times r^2 \times h[/tex]
[tex]V _{cylinder} = A \times h = A \times 6 = 6A[/tex] (2)
From equations (1) and (2) we can say that
The volume of the triangular prism is equal to the volume of the cylinder.
What is a triangular prism?A three-sided polyhedron consisting of a triangle base, a translated copy, and three faces connecting equivalent sides is known as a triangular prism in geometry.If the sides of a right triangular prism are not rectangular, the prism is oblique. Right triangle prisms with square sides and equilateral bases are known as uniform triangle prisms.It is, in essence, a polyhedron with two parallel sides and three surface normals that are all in the same plane (which is not necessarily parallel to the base planes).There are parallelograms in these three faces. The identical triangle appears in every cross-section running parallel to the base faces.To learn more about triangular prism with the given link
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I have a list of five numbers, where the first number is 1 and the fifth number is 5. Each of the other numbers is equal to one more than the average of its two neighbors. What is the sum of all five numbers in the list?
The sum of all five numbers in the list is 25.
According to the question,
The first number on my list of five numbers is 1, and the fifth is 5. The other figures are all one more than their two nearest neighbors' averages.
Let the list of numbers be 1,x,y,z,5.
As each of the other numbers is equal to one more than the average of its two neighbors.
x=(1+y)/2+1
2x=3+y -(1)
Similarly,
y=(x+z)/2+1
2y=x+z+2 -(2)
z=(y+5)/2+1
2z=y+7 -(3)
Multiply equation (2) by 2 and then substitute equations (1) and (3),
4y=3+y+y+7+4
4y=2y+14
2y=14
y=14/2
y=7
From equation (1),
2x=3+7
2x=10
x=5
From equation (3),
2z=7+7
2z=14
z=14/2
z=7
The list is: 1,5,7,7,5
Sum of the numbers of list = 1+5+7+7+5 = 25
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please help!!! i just need this one
Answer:
[tex]p = x \\ r = 6\% \\ t = 2yrs[/tex]
[tex]p = 1600[/tex]
please see the attachment
Answer:
P = $1600
Step-by-step explanation:
We have to use the equation:
[tex]\boxed{I = Prt}[/tex] .
We know that:
• I = $192
• P = ?
• r = 6% = 0.06
• t = 2 years
Substituting these values into the equation:
[tex]192 = P \times 0.06 \times 2[/tex]
Solving for P:
[tex]P = \frac{192}{0.06 \times 2}[/tex]
[tex]P = \bf \$1600[/tex]
What is the perimeter of the given triangle?
10 units
16 units
17.2 units
19.2 units
Answer:
17.2 units
Step-by-step explanation:
You would use the Pythagorean Theorem to solve. The height (a) is 4 units. The base (b) is 6 units.
4^2 + 6^2 = sqrt(52)
This would mean that the hypotenuse is 7.211102550927979 units or 7.2 units (the square root of 4^2 + 6^2).
Now you would need to just add all three sides.
6 + 4 + 7.2 = 17.2 units
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y=alpha sinx +beta cosx is the solution of d^2y/dx^2+dy/dx=0 where (alpha and beta are constant)
The expression y = α · sin x + β · cos x is a solution of the ordinary differential equations if and only if (α, β) = (0, 0).
When a given equation is a solution of an ordinary differential equation
According to the statement, we must find in what conditions a given expression may be a solution of an ordinary differential equation. Then, first and second derivatives of the equation are:
y' = α · cos x - β · sin x (1)
y'' = - α · sin x - β · cos x (2)
Then, we substitute on the ordinary differential equation:
(- α · sin x - β · cos x) + (α · cos x - β · sin x) = 0
And by algebraic handling we simplify the resulting expression:
- (α + β) · sin x + (α - β) · cos x = 0
Where each coefficient represents a constant of a linear combination:
α + β = 0
α - β = 0
Then, the solution of the system of linear equations is (α, β) = (0, 0). The expression y = α · sin x + β · cos x is a solution of the ordinary differential equations if and only if (α, β) = (0, 0).
RemarkThe statement is incomplete and complete form cannot be found, then we decided to create a new statement:
Please prove that y = α · sin x + β · cos x is the solution of the differential equation d²y / dx² + dy /dx = 0 where the following condition is observed, if and only if α = β = 0.
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If a true-false test with 10 questions is given what is the probability of scoring?
Answer:50%
Step-by-step explanation: It doesn't matter how many questions there are because for each question, you have a 50% chance of getting it right. So, the probability is 50%.
What are the solutions of the equation (x 2)2 12(x 2) – 14 = 0? use u substitution and the quadratic formula to solve.
The solutions of the given equation is x = -8 + 5√2 and x = -8 - 5√2.
The quadratic formula is used to find the roots of a quadratic equation. This formula helps to evaluate the solution of quadratic equations replacing the factorization method. If a quadratic equation does not contain real roots, then the quadratic formula helps to find the imaginary roots of that equation. The quadratic formula is also known as Shreedhara Acharya’s formula
(x + 2)² + 12(x + 2) – 14 = 0
Replacing x + 2 with u we get,
u² + 12u - 14
using quadratic formula we get
u = (-12 ±√(12² - 4x1 x-14))/2
u =(-12 ±√200)/2
u = -6 ± 5√2
x +2 = u = -6 ± 5√2
x = u = -8 ± 5√2.
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5. Elias has 1/2 of a candy bar. He shares 1/4 of it with his best friend. How much of the candy bar did his best friend receive?
Answer:
1/8
Step-by-step explanation:
say the bar has 20 pieces. he has 10 pieces. and he gave away 1/4 of those 10
1/4 x 10 = 2.5
20 / 2.5 = 8
So his friend recieved 1/8
you could also just do 1/2 x 1/4 which is 1/8
Answer:
1/8 of the candy bar.
Step-by-step explanation:
Elias has 1/2 of a candy bar.
He gave 1/4 of 1/2 to his friend.
[tex]\sf Friend's s\ share = \dfrac{1}{4} \ of \ \dfrac{1}{2}[/tex]
[tex]\sf =\dfrac{1}{4}*\dfrac{1}{2}\\\\=\dfrac{1}{8}[/tex]
A function has a slope of 3, and one solution is given in the table. identify the missing outputs. x: 5 6 7 8 9 y: -50
The remaining value will increase by the value of the slope. Hence the remaining missing values are -47, -44, -41, -38 and -35
Slope and tablesThe slope of a line is the ratio of the rise to run of a line. It is also defined as the rate of change of coordinate y with respect to x. Mathematically;
slope = change in y/change in x
If the slope of the table given is 3, then using the coordinate points (5, -50)and (6, y)
Substitute
3 = y-(-50)/6-5
3 = y+50/1
y+50 = 3
y = -47
The remaining value will increase by the value of the slope. Hence the remaining missing values are -44, -41, -38 and -35
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Laura completed the following steps to find a product.
Multiply: Three-sevenths times 8
Step 1: 8 times three-sevenths
Step 2: StartFraction (8 plus 3) over 7 EndFraction
Step 3: Eleven-sevenths
Step 4: 1 and four-sevenths
In which step did Laura make her first mistake?
Step 1
Step 2
Step 3
Step 4
The solution to the product of 8 and three-seventh is 2 2/7. According to Laura, she made mistake in step 2 by adding 8 and 3 instead of multiplying
Multiplication of fractions and integers
Fractions are written as a ratio of two integers. For instance a/b is a fraction where a and b are integers.
Given the following equation
Multiply 3/7 and 8
This is expressed mathematically as;
3/7 * 8
Step 1: Swap to have;
8 * 3/7
Step 2: Group the numerator
(8*2)/7
Step 3; Simplify
16/7
Step 4; Convert to mixed fraction
16/7= 2 2/7
The solution to the product of 8 and three-seventh is 2 2/7. According to Laura, she made mistake in step 2 by adding 8 and 3 instead of multiplying
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An open-faced box is to be constructed by cutting squares out of the corners of a 12in by 12in
sheet of tin and folding up the sides. Which of the following describes the volume of such a
box?* (5 Points)
Step-by-step explanation:
Volume= lbh
= (12-2h)(12-2h)h
= 144h-48h²+4h³
The volume of the open-faced box is described by the polynomial equation:Volume = 4x³ - 48x² + 144x
To determine the volume of the open-faced box, we need to consider the dimensions of the box after cutting squares out of the corners and folding up the sides.
Let's assume that each side of the square cutout has a length of x inches.
When the squares are cut out and the sides are folded up, the resulting box will have dimensions:
Length = 12 - 2x inches
Width = 12 - 2x inches
Height = x inches
The volume of the box can be calculated by multiplying these dimensions:
Volume = Length * Width * Height
Volume = (12 - 2x) * (12 - 2x) * x
Volume = 4x³ - 48x² + 144x
Therefore, the volume of the open-faced box is described by the polynomial equation:
Volume = 4x³ - 48x² + 144x
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What is the probability distribution of X when X~B(1,1/25)?
P(X= 0) = 0.96
P(X= 1) = 0.04
P(X= 0) = 0.75
P(X= 1) = 0.25
P(X= 0) = 0.04
P(X= 1) = 0.96
P(X= 0) = 0.25
P(X= 1) = 0.75
The probability distribution of X when X~B(1,1/25) is (Option A)
P(X= 0) = 0.96P(X= 1) = 0.04See the explanation below.
What is the explanation to the above solution?Given X~B(n,p)
P(x=1) = C¹ₙ * P¹ * (1-P)ⁿ⁻¹ (n≥1)
Thus, X~B [1, 1/25]
1/25 = 0.04
hence, p(x=0)
= C⁰₁ * 0.04⁰ (1 - 0.04)¹
= 0.96
P (x=1)
= C¹₁ 0.04¹ (1-0.04)⁰
= 0.04
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How do I get my answer
Factors to zero inverse operations
The zeros of the given equation are -5 and -7
Zeros of a quadratic equationQuadratic equations are equations that has a leading degree of 2. Given the factors of a quadratic equation as expressed below;
(-3x - 15)(x+7) = 0
The expressions -3x -15 and x + 7 are the factors of the equation. Equating both factors to zero
-3x - 15 = 0
Add 15 to both sides of the equation
-3x -15 + 15 = 0 + 15
-3x = 15
Divide both sides of the equation by -3
-3x/-3 = 15/-3
x = -5
Similarly;
x + 7 = 0
x = -7
Hence the zeros of the given equation are -5 and -7
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The difference of two numbers is 3/4. The sum of the two numbers is 9 1/4. Find the number
My car uses 8.5L of petrol per 100km travelled. If petrol costs $2.05 per litre, how much will the petrol cost for my trip?
the petrol cost for the trip is:
C = 2.125*$2.05 = $4.36
How to get the cost of the trip?After a search online, I've found that the trip is of 25km.
Here we know that the car uses 8.5L per 100km, then in 25 km, the car will use one-fourth of 8.5L, that is:
8.5L/4 = 2.125L
So the volume of petrol that the car uses for the trio is 2.125 liters of petrol.
And each liter costs $2.05, then the petrol cost for the trip is given by the product between the total volume of petrol consumed and the cost per liter.
C = 2.125*$2.05 = $4.36
Thus, we conclude that the petrol cost for the trip is 4.36 dollars.
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An experiment consists of drawing 1 card from a standard 52 card deck. let e be the event that the card drawn is a red card. find p(e)
The probability of drawing a red card from a standard deck of cards is 1/2.
Given that there is a standard deck of cards.
We are required to find the probability of drawing a rd card from a standard deck of cards.
Probability is the calculation of chance of happening an event among all the events possible. It lies between 0 and 1. It cannot be negative.
Probability=Number of items/Total items.
Total number of cards=52
Number of red cards =26
Number of black cards=26
Probability of drawing a red card =Number of red cards/ Total cards.
=26/52
=1/2
Hence the probability of drawing a red card from a standard deck of cards is 1/2.
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If (fg)(x) = h(x) such that h of x is equal to the square root of the quantity 8 times x plus 6 end quantity which of the following could accurately represent f and g?
The value of the functions f(x) and g(x) will be √(4x + 3) and √2. Then the correct option is B.
What is a function?A statement, principle, or policy that creates the link between two variables is known as a function. Functions are found all across mathematics and are required for the creation of complex relationships.
If (f g)(x) = h(x) such that h(x) = √(8x + 6). Then we have
(f g)(x) = h(x)
f(x) · g(x) = h(x)
Then put the value of h(x), then we have
f(x) · g(x) = √(8x + 6)
f(x) · g(x) = √2(4x + 3)
f(x) · g(x) = √(4x + 3) × √2
Thus, the value of the functions f(x) and g(x) will be √(4x + 3) and √2.
Then the correct option is B.
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