Select the first correct reason why the given series converges.


a. convergent geometric series


b. convergent p-series


c. integral test


d. comparison with a convergent p-series


e. converges by limit comparison test f. converges by alternating series test e 1. ∑n=2∞n2+n‾√n4−4 f 2. ∑n=1∞n2+ln(n)n2−9n f 3. ∑n=1∞(cos(nπ))ln(6n) f 4. ∑n=1∞(−1)n7n+5 f 5. ∑n=1∞(−1)nn‾√n+4 c 6. ∑n=2∞4n(ln(n))2

Answer:

Step-by-step explanation:

The outcome of the game is

Toss 1               Toss 2      Result

H                         H               18

H                         T                 -1

T                         H                -1

T                         T                -20

His only winning position is H ---- H

That means he has only a 1/4 chance in winning. He shouldn't play at all. It might be a fair coin, but it is not a fair game.

➷ The formula for a right triangle is:

(b x h) / 2

We know this:

(12 x n) / 2 = 30

Multiply both sides by 2:

12 x n = 60

Divide both sides by 12:

n = 5

Now we know the height is 5

We can now use Pythagoras' theorem to calculate the hypotenuse:

c^2 = 5^2 + 12^2

c^2 = 169

Square root it:

c = 13

The hypotenuse is 13 ft

➶ Hope This Helps You!

➶ Good Luck (:

➶ Have A Great Day ^-^

↬ ʜᴀɴɴᴀʜ ♡

Answer:

The formula for working out the area of a triangle:

(b x h) / 2

[tex](12*h)/2=30\\(*2)\\12*h=60h\\h=5[/tex]

Pythagoras' theorem to calculate the hypotenuse:

c² = 5² + 12²

c² = 169

c = 13

The hypotenuse is 13 ft

Step-by-step explanation:

Answer:

log9(11x)

Step-by-step explanation:

log9(11) + log9(x)

We know that loga(b) + loga(c) = loga(b*c)

log9(11*x)

log9(11x)

Final answer:

To rewrite a sum of two logarithms as a single logarithm, combine them using the property that the log of a product equals the sum of the logs. For example, log a + log b becomes log (ab).

Explanation:

To rewrite the sum of two logarithms as a single logarithm, we use the relevant log properties. One of the most important properties of exponents is that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers, which is log xy = log x + log y. Conversely, the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers. So, if you have two logarithms of the same base you're adding, like log a + log b, you can combine them into a single log by multiplying the two arguments to get log (a*b).

An example using numbers could look like this:

log₂(3) + log₂(5) = log₂(3*5) = log₂(15)

Therefore, the sum of two logarithms, log a + log b, can be rewritten as log (ab), which expresses the multiplication of a and b within a single logarithm.

Answer:

x     y

-2   7

-1    10.5

0    15.75

1     23.625

2    35.4375

Step-by-step explanation:

The general equation of the exponential function is [tex]y=ab^x[/tex].

We know from our table that when [tex]x=0[/tex], [tex]y=15.75[/tex]. Let's replace those values in our equation:

[tex]y=ab^x[/tex]

[tex]15.75=ab^0[/tex]

Remember that [tex]b^0=1[/tex], so:

[tex]15.75=a(1)[/tex]

[tex]15.75=a[/tex]

[tex]a=15.75[/tex]

We also know from our table that when [tex]x=-1[/tex], [tex]y=10.5[/tex]. Let's replace the values again:

[tex]y=ab^x[/tex]

[tex]10.5=ab^{-1}[/tex]

But we now know that [tex]a=15.75[/tex], so let's replace that value as well:

[tex]10.5=15.75b^{-1}[/tex]

Remember that [tex]b^{-1}=\frac{1}{b}[/tex], so:

[tex]10.5=\frac{15.75}{b}[/tex]

[tex]10.5b=15.75[/tex]

[tex]b=\frac{15.75}{10.5}[/tex]

[tex]b=1.5[/tex]

Now, we can put it all together to complete our exponential function:

[tex]y=ab^x[/tex]

[tex]y=15.75(1.5)^x[/tex]

To find the missing values, we just need to evaluate our function at [tex]x=1[/tex] and [tex]x=2[/tex]:

- For [tex]x=1[/tex]

[tex]y=15.75(1.5)^x[/tex]

[tex]y=15.75(1.5)^1[/tex]

[tex]y=23.625[/tex]

- For [tex]x=2[/tex]

[tex]y=15.75(1.5)^x[/tex]

[tex]y=15.75(1.5)^2[/tex]

[tex]y=35.4375[/tex]

Answer:

Original price of the dress was $97.88

Step-by-step explanation:

40/100=39.15/d

0.4d=39.15

0.4d=39.15

d=39.15/0.4

d=97.88

The original price of the dress is $97.875.

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.

Variables are the name given to these symbols because they lack set values.

In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.

Given:

Danielle paid $39.15.

But Danielle paid after 40% off.

So, the price before discount

40% of x = 39.15

x= 3915/40

x= $97.875

Hence, the original price is $97.875.

Learn more about Algebra here:

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Answer:

[tex]3,358\ people[/tex]  

Step-by-step explanation:

The formula to calculate the exponential growth function is equal to

[tex]f(x)=P(e)^{rx}[/tex]  

where  

f(x) is the population  

P is the population in the year 2000  

r is the rate in decimal  

x is number of years since 2000  

e is the mathematical constant number

we have  

[tex]x=2020-2000=20\ years\\ P=2,400\\ r=0.0168[/tex]  

substitute in the formula above  

[tex]f(x)=2,400(e)^{0.0168*20}[/tex]  

[tex]f(x)=3,358\ people[/tex]  

Answer:

The answer is circle; (x')² + (y')² - 4 = 0

Step-by-step explanation:

* At first lets talk about the general form of the conic equation

- Ax² + Bxy + Cy²  + Dx + Ey + F = 0

∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.

∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.

∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.

* Now we will study our equation:

* 2x² + 2y² = 8

∵ A = 2 , B = 0 , C = 2

∴ B² - 4AC = (0) - 4(2)(2) = -16 < 0

∵ B² - 4AC < 0

∴  it will be either a circle or an ellipse

* Lets use this note to chose the correct figure

- If A and C are equal and nonzero and have the same sign,

 then the graph is a circle.

- If A and C are nonzero, have the same sign, and are not equal

 to each other, then the graph is an ellipse.

∵ A = 2 and C = 2

∴ The graph is a circle.

∵ D and E = 0

∴ The center of the circle is the origin (0 , 0)

∵ Ф = 30°

∴ The point (x , y) will be (x' , y')

- Where x = x'cosФ - y' sinФ and y = x'sinФ + y'cosФ

∴ x = x'cos(30°) - y'sin(30°)

∴ y = x'sin(30°) + y'cos(30°)

∴ x = (√3/2)x' - (1/2)y' and y = (1/2)x' + (√3/2)y'

∴ [tex]x=\frac{\sqrt{3}x'-y'}{2}[/tex]

∴ [tex]y=\frac{x'+\sqrt{3}y'}{2}[/tex]

* Lets substitute x and y in the first equation

∴ [tex]2(\frac{\sqrt{3}x'-y'}{2})^{2}+2(\frac{x'+\sqrt{3}y'}{2})^{2}=8[/tex]

* Use the foil method

∴ [tex]2(\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}}{4})+2(\frac{x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{4})=8[/tex]

* Open the brackets

∴ [tex]\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}+x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{2}=8[/tex]

* Collect the like terms

∴ [tex]\frac{4x'^{2}+4y'^{2}}{2}=8[/tex]

* Simplify the fraction

∴ 2(x')² + 2(y')²= 8

* Divide each side by 2

∴ (x')² + (y')² = 4

∴ The equation of the circle is (x')² + (y')² = 4

* The general equation of the circle is (x')² + (y')² - 4 = 0  

 after rotation 30° about the origin

* Look to the graph

- The blue circle for the equation 2x² + 2y² = 8

- The blue circle for equation (x')² + (y')² - 4 = 0

* That is because the two circles have same centers and radii

- The green line is x' and the purple line is y'

Answer:

The answer is D

Good luck on the Ed-genuity test

BCA is 90 degrees, because a tangent line is perpendicular to the raduis line BC.

Final answer:

The measure of angle M is 90 degrees.

Explanation:

The measure of angle M is 90 degrees. To understand why, we can use the fact that a line tangent to a circle is perpendicular to the radius drawn to the point of tangency. In this case, line AD is tangent to circle B at point C, so we can draw line AC, which is the radius of the circle, and find the measure of angle M by finding the complement of the angle formed by lines AD and AC. Since the complement of a 90-degree angle is 90 degrees, the measure of angle M is 90 degrees.

ANSWER

[tex] \angle \: KGI =50 \degree[/tex]

[tex]\angle \: KJI = 130 \degree[/tex]

EXPLANATION

Angles in the same segment are equal.

This implies that;

[tex] \angle \: KGI = \angle \:H[/tex]

Hence,

[tex] \angle \: KGI =50 \degree[/tex]

Also GJKI is a cyclic quadrilateral.

The opposite angles of a cyclic quadrilateral sums up to 180°.

This means that,

[tex] \angle \: KJI + \angle \: KGI = 180 \degree[/tex]

[tex] \angle \: KJI +50 \degree = 180 \degree[/tex]

[tex]\angle \: KJI = 180 \degree - 50 \degree[/tex]

[tex]\angle \: KJI = 130 \degree[/tex]

Answer:

b. [tex]\csc(x)[/tex]

Step-by-step explanation:

The given expression is

[tex]\frac{\sec(x)}{\tan(x)}[/tex]

We express in terms of basic trigonometric ratios to obtain;

[tex]\frac{\frac{1}{\cos(x)} }{\frac{\sin(x)}{\cos(x)} }[/tex]

This is the same as

[tex]\frac{1}{\cos(x)}\div \frac{\sin(x)}{\cos(x)}[/tex]

[tex]\frac{1}{\cos(x)}\times \frac{\cos(x)}{\sin(x)}[/tex]

Cancel out the common factors;

[tex]\frac{1}{\sin(x)}=\csc(x)[/tex]

Answer:

[tex]\frac{secx}{tanx}[/tex]  = cscx

Step-by-step explanation:

We have given a trigonometric expression.

[tex]\frac{secx}{tanx}[/tex]

We have to simplify the above expression.

Since, we know that

secx is reciprocal of cosx.

secx  =  1/cosx

Tanx is the ratio of sinx and cosx.

Tanx  =  sinx / cosx

Given expression becomes

[tex]\frac{1/cosx}{sinx/cosx}[/tex]

[tex]\frac{1}{cosx}\frac{cosx}{sinx}[/tex]

[tex]\frac{1}{sinx}[/tex]

[tex]\frac{secx}{tanx}[/tex]  = cscx which is the answer.

Answer:

2.5 %  

Step-by-step explanation:

The simple interest formula is

I = Prt

Data:

I = $75

P = $600

t = 5 yr

Calculation:

75 = 600 × r × 5

75 = 3000 r

Divide each side by 3000

r = 75/3000 = 0.025 = 2.5 % APR

The annual percentage rate is 2.5 %.

Final answer:

The annual rate of simple interest that Remi received from his savings account is 2.5%. This was calculated using the simple interest formula I = PRT, rearranged to solve for R, the annual interest rate.

Explanation:

To work out the annual rate of simple interest that Remi received from his savings account, we can use the formula for simple interest:

I = PRT

where I is the total interest earned, P is the principal amount invested, R is the annual interest rate, and T is the time the money is invested in years.

From the question, we know that Remi invested £600 (P=600), received £75 in interest (I=75), and the investment was for 5 years (T=5). We need to find the annual rate (R).

Rearranging the formula to get value of R, we get:

R = I / (PT)

Substituting the given values:

R = 75 / (600 × 5) (× is the multiplication symbol)

R = 75 / 3000

R = 0.025

To Convert decimal to a percentage, we multiply by 100:

R = 0.025 × 100

R = 2.5%

So, the annual rate of simple interest that Remi received is 2.5%.

Answer:

Step-by-step explanation:

Your car breaks down unexpectedly and you must have it repaired immediately.  That's an example of a situation where you might use money from a financial reserve.

Answer:

see explanation

Step-by-step explanation:

(1)

given

4x² = 81 ( divide both sides by 4 )

x² = [tex]\frac{81}{4}[/tex] ( take the square root of both sides )

x = ± [tex]\sqrt{\frac{81}{4} }[/tex] = ± [tex]\frac{9}{2}[/tex]

---------------------------------------------------------------------

(2)

subtract 7 from both sides

(4x - 3)² = 32 ( take the square root of both sides )

4x - 3 = ± [tex]\sqrt{32}[/tex] = ± 4[tex]\sqrt{2}[/tex]

Add 3 to both sides

4x = 3 ± 4[tex]\sqrt{2}[/tex] ( divide both sides by 4 )

x = [tex]\frac{3}{4}[/tex] ± [tex]\sqrt{2}[/tex]

Answer:

B - closing costs are increased

Step-by-step explanation:

All of the other options are advantages to purchasing points. You want lower monthly payments (a), a tax break (c), and lower interest rate (d)

Answer:                

The correct answer is option B.closing costs are increased.

Step-by-step explanation:                        

Mortgage points or discount points refers to the fees paid directly to the lender during closing in exchange for a reduced interest rate.

In this way the buyer buys down the rate, that helps in reducing his monthly mortgage payments.

So, option A, C and D are the advantages of purchasing discount points.

Answer:

Final answer is [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex].

Step-by-step explanation:

Given equation is [tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex]

Now we need to find the solution of  [tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex] in given interval [tex][0, 2\pi ][/tex].

[tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex]

[tex]-\cos\left(\theta\right)=\frac{1}{2}-1[/tex]

[tex]-\cos\left(\theta\right)=-\frac{1}{2}[/tex]

[tex]\cos\left(\theta\right)=\frac{1}{2}[/tex]

which gives [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex] in the given interval.

Hence final answer is [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex].

In the interval [0, 2π], the cosine function takes on the value 1/2 at two specific angles: π/3 and 5π/3.

To solve the equation 1 - cos(theta) = 1/2 on the interval [0, 2π], this is done by:

Subtract 1/2 from both sides of the equation to isolate the cosine term:

1 - cos(theta) - 1/2 = 0

-cos(theta) + 1/2 = 0

So multiply both sides by -1 to get rid of the negative sign: cos(theta) - 1/2 = 0

1/2 can be written as 2/4: cos(theta) - 2/4 = 0

So look for common denominator for the fraction on the left side, and is 4:

(4*cos(theta) - 2)/4 = 0

Then Multiply both sides by 4 to remove the fraction: 4*cos(theta) - 2 = 0

So, add 2 to both sides: 4*cos(theta) = 2

Lastly divide both sides by 4: cos(theta) = 1/2

Therefore, the solutions to the equation are: theta = π/3 and theta = 5π/3.

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Answer:

your answer is x=16/11

explanation:

solve for x by simplifying both sides of the equation then isolation the variable.

exact form: 16/11

decimal form: 1.45

mixed number form: 1  5/11

hope this helps

~~bangtanboys7

Answer: [tex]x=\frac{16}{11}[/tex]

Step-by-step explanation:

To solve the equation shown in the problem you must:

- Add the like terms, as following:

[tex]\frac{3}{4}x+2x=\frac{5}{3}+\frac{7}{3}\\ \frac{11}{4}x=4[/tex]

- Multiply both sides by 4.

- Divide both sides of the equation by 11.

Therefore, the result is:

 [tex](4)\frac{11}{4}x=4(4)\\11x=16\\x=\frac{16}{11}[/tex]

Answer:

S3 = 39

Step-by-step explanation:

* an = 3(3)^(n-1) is a geometric sequence

* The general rule of the geometric sequence is:

 an = a(r)^(n-1)

Where:

a is the first term

r is the common difference between each consecutive terms

n is the position of the term in the sequence

The rules means:

- a1 = a , a2 = ar , a3 = ar² , a4 = ar³ , ........................

∵ an = 3(3)^(n-1)

∴ a = 3 and r = 3

∴ a1 = 3

∴ a2 = 3(3) = 9

∴ a3 = 3(3)² = 27

* S3 = a1 + a2 + a3

∴ S3 = 3 + 9 + 27 = 39

Note:

We can use the rule of the sum:

Sn = a(1 - r^n)/(1 - r)

S3 = 3(1 - 3³)/1 - 3 = 3(1 - 27)/-2 = 3(-26)/-2 =3(13) = 39

Answer:

The correct answer is S₃  = 39

Step-by-step explanation:

It is given that,

aₙ = 3(3)ⁿ⁻¹

To find a₁

a₁ = 3(3)¹⁻¹ = 3(3)°

= 3 * 1 = 3

To find a₂

a₂ = 3(3)²⁻¹ = 3(3)¹

= 3 * 3 = 9

To find a₃

a₃ = 3(3)³⁻¹ = 3(3)²

= 3 * 9 = 27

To find the value of S₃

S₃ = a₁ + a₂ + a₃

 = 3 + 9 + 27 = 39

Therefore the correct answer is S₃  = 39

$240

800x.06= 48

48x5=240

Answer:

The answer is 9.6

hope this helps!

Answer:

Step-by-step explanation:

The ratios of corresponding linear dimensions are identical:

  5/2 = 15/6 = 2.5

so we can conclude the figures are similar. The scale factor is the ratio of corresponding dimensions:

  first : second = 5 : 2 . . . or . . . 2.5 : 1

Answer:

Yes.

Scale factor = 2.5.

Step-by-step explanation:

They are similar because corresponding dimensions are in the same ratio:

2/5 = 6/15.

The scale factor = 5/2 = 2.5.

Answer:

The answer is 1/2 which i think that's what you meant from C.

Step-by-step explanation:

A standard deck of cards has 52 cards. 13 of these cards are spades and 13 are diamonds.

the answer to your question is c :]

Final answer:

The division algorithm for polynomials establishes that a polynomial p(x) can be divided by a non-zero polynomial d(x) (with degree less or equal to p(x)), to yield unique quotient q(x) and remainder r(x), with r(x) having a lower degree than d(x).

Explanation:

The division algorithm in the context of polynomials is a fundamental concept in algebra that stipulates for any two polynomials p(x) and d(x), with d(x) ≠ 0 and the degree of d(x) less than or equal to the degree of p(x), there exist unique polynomials q(x) and r(x) such that p(x) = d(x) × q(x) + r(x).

In this scenario, q(x) is referred to as the quotient and r(x) is the remainder. The degree of the remainder r(x) will always be less than the degree of d(x), following the division algorithm.

When applying this algorithm to special sets of polynomials like Legendre polynomials, additional properties can be observed, such as the roots of the polynomials or specific transformations like the Poisson bracket that could arise in mathematical physics. Moreover, the concept extends to rational functions, which are the quotients of polynomials.

Answer:

x = log base 4 of 2

Step-by-step explanation:

I'm assuming this is truly an exponential function as you state and not a mistype.

6^x = 2

Take the log of each side:

x · log(6) = log(2)

Divide each side by log(6):

x = log(2) / log(6)

x = 0.30103 / 0.77815

x = 0.3869... (rounded)

Answer:

0.0085 centimeters.

Step-by-step explanation:

85/10000 is equal to 0.0085 centimeters, which is absurdly thin for anything, really.

a) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. b) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. c) Yes, s is a subspace of v because the three conditions for a subspace are satisfied.

a. v=mn(r) is the vector space of all m x n matrices with entries in the real numbers. In this case, s is a subspace of v if it satisfies the three conditions: (1) the zero vector is in s, (2) s is closed under vector addition, and (3) s is closed under scalar multiplication. Since the zero matrix is an upper triangular matrix, it satisfies the first condition. If two upper triangular matrices are added, the result will also be an upper triangular matrix, satisfying the second condition. And if an upper triangular matrix is multiplied by a scalar, the result is still an upper triangular matrix, satisfying the third condition. Therefore, s is a subspace of v.

b. In this case, v is the vector space of all real-valued functions defined on the interval (-∞, ∞). The subset s consists of functions satisfying f(0) = 0. Similar to part a, we need to check if s satisfies the three conditions to be considered a subspace. The zero function satisfies f(0) = 0, so it satisfies the first condition. If two functions in s are added, the sum will also have f(0) = 0, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still have f(0) = 0, satisfying the third condition. Therefore, s is a subspace of v.

c. In this case, v is the vector space of all second-order linear homogeneous differential equations. The subset s consists of functions satisfying y'' - 4y' + 3y = 0. To be considered a subspace, s would need to satisfy the three conditions: (1) the zero element is in s, (2) s is closed under addition, and (3) s is closed under scalar multiplication. The zero function satisfies the differential equation, so it satisfies the first condition. If two functions in s are added, the sum will also satisfy the differential equation, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still satisfy the differential equation, satisfying the third condition. Therefore, s is a subspace of v.

Answer:

Step-by-step explanation:

Look at the picture.

A₁ = 2 · 2 = 4

A₂ = 2 · 3 = 6

The area of a rhombus:

A = A₁ + A₂ ⇒ A = 4 + 6 = 10

You can use the formula of an area of a rhombus:

[tex]A=\dfrac{d_1d_2}{2}[/tex]

d₁ = 4, d₂ = 5

Substitute:

[tex]A=\dfrac{(4)(5)}{2}=(2)(5)=10[/tex]

Answer:

10 ft²

Step-by-step explanation:

A = 1/2 (d1) (d2)

A = 1/2 (4) (5)

1/2 x 20

=10 ft²

Answer:

17.85 months

Step-by-step explanation:

original number of rabbits=50

% increase per month=12%

number of months=x

form equation for 1000 rabbits

(112/100) ×50× x = 1000

56x=1000

x=1000/56

x=17.857 months

Answer:

I would say about 60 to 70 pounds. But I have no more information from you to better answer your question, so that's all I have right now.

Step-by-step explanation:

Answer:

the answer is D. 3^7√3

Step-by-step explanation:

For this case we have to define properties of powers and roots:

[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]

We also have to:

[tex]3 ^ {15} = 3 ^ {14} * 3[/tex]

So:

[tex]\sqrt {3 ^ {15}} = \sqrt {3 ^ {14} * 3} = 3 ^ {\frac {14} {2}} * \sqrt {3} = 3 ^ 7 * \sqrt {3}[/tex]

Answer:

[tex]3 ^ 7 * \sqrt {3}[/tex]

Option D