3. For each triangle, find the length of the labeled side.

Answer:

Hope the picture will help you

The sum we want is

[tex]\displaystyle \sum_{n=0}^\infty \frac{(-1)^{T_n}}{(2n+1)^2} = 1 - \frac1{3^2} - \frac1{5^2} + \frac1{7^2} + \cdots[/tex]

where [tex]T_n=\frac{n(n+1)}2[/tex] is the n-th triangular number, with a repeating sign pattern (+, -, -, +). We can rewrite this sum as

[tex]\displaystyle \sum_{k=0}^\infty \left(\frac1{(8k+1)^2} - \frac1{(8k+3)^2} - \frac1{(8k+7)^2} + \frac1{(8k+7)^2}\right)[/tex]

For convenience, I'll use the abbreviations

[tex]S_m = \displaystyle \sum_{k=0}^\infty \frac1{(8k+m)^2}[/tex]

[tex]{S_m}' = \displaystyle \sum_{k=0}^\infty \frac{(-1)^k}{(8k+m)^2}[/tex]

for m ∈ {1, 2, 3, …, 7}, as well as the well-known series

[tex]\displaystyle \sum_{k=1}^\infty \frac{(-1)^k}{k^2} = -\frac{\pi^2}{12}[/tex]

We want to find [tex]S_1-S_3-S_5+S_7[/tex].

Consider the periodic function [tex]f(x) = \left(x-\frac12\right)^2[/tex] on the interval [0, 1], which has the Fourier expansion

[tex]f(x) = \frac1{12} + \frac1{\pi^2} \sum_{n=1}^\infty \frac{\cos(2\pi nx)}{n^2}[/tex]

That is, since f(x) is even,

[tex]f(x) = a_0 + \displaystyle \sum_{n=1}^\infty a_n \cos(2\pi nx)[/tex]

where

[tex]a_0 = \displaystyle \int_0^1 f(x) \, dx = \frac1{12}[/tex]

[tex]a_n = \displaystyle 2 \int_0^1 f(x) \cos(2\pi nx) \, dx = \frac1{n^2\pi^2}[/tex]

(See attached for a plot of f(x) along with its Fourier expansion up to order n = 10.)

Expand the Fourier series to get sums resembling the [tex]S'[/tex]-s :

[tex]\displaystyle f(x) = \frac1{12} + \frac1{\pi^2} \left(\sum_{k=0}^\infty \frac{\cos(2\pi(8k+1) x)}{(8k+1)^2} + \sum_{k=0}^\infty \frac{\cos(2\pi(8k+2) x)}{(8k+2)^2} + \cdots \right. \\ \,\,\,\, \left. + \sum_{k=0}^\infty \frac{\cos(2\pi(8k+7) x)}{(8k+7)^2} + \sum_{k=1}^\infty \frac{\cos(2\pi(8k) x)}{(8k)^2}\right)[/tex]

which reduces to the identity

[tex]\pi^2\left(\left(x-\dfrac12\right)^2-\dfrac{21}{256}\right) = \\\\ \cos(2\pi x) {S_1}' + \cos(4\pi x) {S_2}' + \cos(6\pi x) {S_3}' + \cos(8\pi x) {S_4}' \\\\ \,\,\,\, + \cos(10\pi x) {S_5}' + \cos(12\pi x) {S_6}' + \cos(14\pi x) {S_7}'[/tex]

Evaluating both sides at x for x ∈ {1/8, 3/8, 5/8, 7/8} and solving the system of equations yields the dependent solution

[tex]\begin{cases}{S_4}' = \dfrac{\pi^2}{256} \\\\ {S_1}' - {S_3}' - {S_5}' + {S_7}' = \dfrac{\pi^2}{8\sqrt 2}\end{cases}[/tex]

It turns out that

[tex]{S_1}' - {S_3}' - {S_5}' + {S_7}' = S_1 - S_3 - S_5 + S_7[/tex]

so we're done, and the sum's value is [tex]\boxed{\dfrac{\pi^2}{8\sqrt2}}[/tex].

Answer:

Step-by-step explanation:

Answer:

f(x)=x(x+4)(x+3)(x-4)(x-3)

Step-by-step explanation:

hello :

an  polynomial is : f(x)=(x+4)(x+3)(x-0)(x-4)(x-3)

means : f(x)=x(x+4)(x+3)(x-4)(x-3)

Step-by-step explanation:

what is that called into the comet or ok

Answer:

2x = 26, x = 13

Step-by-step explanation:

Write and solve an equation twice a number is 26

x = unknown number

Equation: 2x = 26

Solve:

2x = 26

/2    /2 <== divide both sides by 2

x = 13

Check your answer:

2x = 26

2(13) = 26

26 = 26

This stament is correct

Hope this helps!

Answer:

A

Step-by-step explanation:

Answer:

x ≥ 4

Step-by-step explanation:

Given:

[tex]\displaystyle \large{3x-1\geq 11}[/tex]

Add both sides by 1:

[tex]\displaystyle \large{3x-1+1\geq 11+1}\\\displaystyle \large{3x\geq 12}[/tex]

Divide both sides by 3:

[tex]\displaystyle \large{\dfrac{3x}{3} \geq \dfrac{12}{3}}\\\displaystyle \large{x\geq 4}[/tex]

Hence, the solution is x ≥ 4

Answer: HELLO THERE!  Here's what I found from study dot  com            "4 divided by 5 is equal to 0.8. This decimal can also be written as a fraction. 0.8 = eight tenths or 8/10 (4/5 in its reduced form)."

Step-by-step explanation:

Hope this helps, have a good day!

[tex]=================================[/tex]

Option A is correct & here's why.

When dividing fractions, we take the first fraction and multiply it times the reciprocal of the second fraction (the second fraction flopped over)

So if we have 1/4, then its reciprocal is 4/1, or just 4.

Hence, Option A (1/5÷4) is correct.

[tex]==============================[/tex]

~DiamonD~

Answer:

88 + 30[tex]\sqrt{7} \\[/tex]

Step-by-step explanation:

Expand to make easier:

(3[tex]\sqrt{7} \\[/tex] + 5) x (3[tex]\sqrt{7} \\[/tex] + 5)

FOIL:

9(7) + 15    [tex]\sqrt{7}[/tex] + 15[tex]\sqrt{7}[/tex] + 25

Simplify:

63 + 30[tex]\sqrt{7}[/tex] + 25

88 + 30[tex]\sqrt{7}[/tex]

Since no diagram attached there are two possible options.

Option 1

The quadrilateral is parallelogram.

In this case ∠A and ∠D sum up to 180°, therefore:

Option 2

The quadrilateral is isosceles trapezoid.

In this case ∠A is congruent with ∠D, therefore:

Answer:

115

Step-by-step explanation:

Answer:  Difference between 21 1/4 - 18 2/4 =2 3/4

Step-by-step explanation:

Answer:

tan m‹G = 28.2/45.8

m‹G= 31.62°

Answer:

196 square feet

Step-by-step explanation:

(learn it before u submit. im here to help not cheat;w;
-ur fellow classmate)

S=ph+2b
s=(p=l+l+w+w) (h)+ (2) (b=lxw)
s=(8+8+10+10) (1) + (2) (8x10)
s=( 36x1 )+ (2x80)
s= 36+160
s=196

God bless ya:)

The area of this prism is 196 square feet.

The prism's area can be calculated by adding its lateral area to the base areas. The process of calculating these areas ends up being facilitated because the two bases of a prism are the same, therefore, it is enough to calculate the area of ​​a base and multiply the result by 2

The formula for the area of ​​the prism is given by:

[tex]S=ph+2b[/tex]

Where:

So we have that:

[tex]s=(p=l+l+w+w) (h)+ (2) (b=lxw)\\s=(8+8+10+10) (1) + (2) (8*10)\\s=( 36*1 )+ (2*80)\\s= 36+160\\s=196[/tex]

See more about area at brainly.com/question/11952845

========================================================

Work Shown:

[tex]n = \hat{p}*(1-\hat{p})\left(\frac{z}{E}\right)^2\\\\n \approx 0.5*(1-0.5)\left(\frac{1.96}{0.03}\right)^2\\\\n \approx 1067.111\\\\n \approx \boldsymbol{1068}\\\\[/tex]

Notes:

Answer:

The answer is 72

Step-by-step explanation:

because you take the 8 times it by the 9

9×8=72 square³

Answer:

no

Step-by-step

Use the slope-intercept form to find the slope and y-intercept.

Tap for fewer steps...

The slope-intercept form is

y

=

m

x

+

b

, where

m

is the slope and

b

is the y-intercept.

y

=

m

x

+

b

Find the values of

m

and

b

using the form

y

=

m

x

+

b

.

m

=

4

b

=

−

6

The slope of the line is the value of

m

, and the y-intercept is the value of

b

.

Slope:

4

y-intercept:

(

0

,

−

6

)

Any line can be graphed using two points. Select two

x

values, and plug them into the equation to find the corresponding

y

values.

Tap for fewer steps...

Find the x-intercept.

Tap for fewer steps...

To find the x-intercept(s), substitute in

0

for

y

and solve for

x

.

0

=

4

x

−

6

Solve the equation.

Tap for fewer steps...

Rewrite the equation as

4

x

−

6

=

0

.

4

x

−

6

=

0

Add

6

to both sides of the equation.

4

x

=

6

Divide each term by

4

and simplify.

Tap for fewer steps...

Divide each term in

4

x

=

6

by

4

.

4

x

4

=

6

4

Cancel the common factor of

4

.

Tap for fewer steps...

Cancel the common factor.

4

x

4

=

6

4

Divide

x

by

1

.

x

=

6

4

Cancel the common factor of

6

and

4

.

Tap for fewer steps...

Factor

2

out of

6

.

x

=

2

(

3

)

4

Cancel the common factors.

Tap for fewer steps...

Factor

2

out of

4

.

x

=

2

⋅

3

2

⋅

2

Cancel the common factor.

x

=

2

⋅

3

2

⋅

2

Rewrite the expression.

x

=

3

2

x-intercept(s) in point form.

x-intercept(s):

(

3

2

,

0

)

Find the y-intercept.

Tap for fewer steps...

To find the y-intercept(s), substitute in

0

for

x

and solve for

y

.

y

=

4

(

0

)

−

6

Solve the equation.

Tap for fewer steps...

Remove parentheses.

y

=

4

(

0

)

−

6

Simplify

4

(

0

)

−

6

.

Tap for fewer steps...

Multiply

4

by

0

.

y

=

0

−

6

Subtract

6

from

0

.

y

=

−

6

y-intercept(s) in point form.

y-intercept(s):

(

0

,

−

6

)

Create a table of the

x

and

y

values.

x

y

0

−

6

3

2

0

Graph the line using the slope and the y-intercept, or the points.

Slope:

4

y-intercept:

(

0

,

−

6

)

x

y

0

−

6

3

2

0

Answer: 35

Step-by-step explanation:

     The average can be found by adding their ages together and dividing by the number of ages.

     Given:

[tex]\frac{19+20+20+21+x}{5}[/tex] = 23

     Multiply both sides of the equation by 5:

19 + 20 + 20 + 21 + x = 115

     Combine like terms:

80 + x = 115

     Subtract 80 from both sides of the equation:

x = 35

                    Erica is 35.

Answer:

solution 1= 40ml

solution 2= 160ml

Step-by-step explanation:

%alcohol= amount of alcohol/total solutionx100

0.52x200=104ml of alcohol present

0.2x L=0.2Lml of alcohol in solution one

0.6 x M=0.6Mml of alcohol in solution two

1st equation

0.2L+0.6M=104ml alcohol

times 10 to get whole

2L+6M=1040ml

2nd

same for this equation

10L+10M=2000ml

10L+10M=2000

2L+6M=1040

elimination method

=20L+20M=4000

-

=20L+60M=10400

-40M=-6400

M=160ml

2L+6 (160)=1040

2L=1040-960

2L=80

L=40ml

The approximate diameter of the rug with an area of 40 ft² is 7.14 feet.

Area is the amount of space occupied by a two dimensional shape or object.

The area of a circle is given by:

Area = π * diameter²/4

The rug has an area of 40 square feet. Hence:

40 = π * diameter²/4

Diameter = 7.14 feet

The approximate diameter of the rug with an area of 40 ft² is 7.14 feet.

Find out more on area at: https://brainly.com/question/25292087

Answer:

45°

Step-by-step explanation:

complementary angles add up to 90° so it's 90-45

Answer: y = 4.5

scale factor = 4/3

Answer:

it depends

Step-by-step explanation:

sometimes a teacher will accept "there are infinite solutions" but if  you cant you may need to do something else(this applies to basically everything)

for an example lets use x=1

(1,+infinity)

or

(1,-infinity)

(this is being graphed.)  (this may not apply to everything)

i hope this helps

Answer:

a joke and a cartoon together cost $80

c=$15   j=$65

Step-by-step explanation:

j=joke

c=cartoon

8j +6c = 610

6j+8c = 510

    Multiply the first by 8

    Multiply the second by -6

64j +48c = 4880

-36j -48c = -3060

      ADD both (c will cancel)

28j = 1820

j= 65

       PLUG j

8(65) + 6c = 610

520+6c = 610

6c= 90

 c = 15

       j + c   = 65+15 = 80

Hello!

x + y = 9,

x=  9-y

x - y + 3z = 2

x= 2+y -3z

4y - 32 - 5 = 0.​

4y = 32+5

4y = 37

y = 37/4

Answer:

⠀

Step-by-step explanation:

⠀

We know,

[tex] {\longrightarrow \qquad \boldsymbol{\pmb{Total \: \: surface \: \: area_{(cube)} = 6a {}^{2} }}}[/tex]

⠀

Where,

⠀

Now, Substituting the values in the formula :

⠀

[tex]{\longrightarrow \qquad \rm{{Total \: \: surface \: \: area_{(cube)} = \sf 6 \times( 2.5) {}^{2} }}}[/tex]

⠀

[tex]{\longrightarrow \qquad \rm{{Total \: \: surface \: \: area_{(cube)} = \sf 6 \times6.25 }}}[/tex]

⠀

[tex]{\longrightarrow \qquad \boldsymbol{ \pmb{Total \: \: surface \: \: area_{(cube)} = \it 37.5 }}}[/tex]

⠀

Therefore,

Answer:

37.5 cm²

Step-by-step explanation:

Given:-

To find:-

Solution:-

Total surface area of cube :- 6a²

6(2.5)² sq. cm

37.5 cm²

Answer:

4 ,-5

Step-by-step explanation: