Find the value of [tex]$6+\frac{1}{2+\frac{1}{6+\frac{1}{2+\frac{1}{6+\cdots}}}}$.[/tex] Your answer will be of the form [tex]$a+b\sqrt{c}$[/tex] where no factor of $c$ (other than $1$) is a square. Find $a+b+c$.

Answers

Answer 1

Let

[tex]x = 6 + \dfrac1{2 + \dfrac1{6 + \dfrac1{2 + \cdots}}}[/tex]

Then

[tex]x = 6 + \dfrac1{2 + \dfrac1x}[/tex]

and solving for x gives

[tex]x = 6 + \dfrac x{2x + 1}[/tex]

[tex]x = \dfrac{13x + 6}{2x + 1}[/tex]

[tex]x(2x+1) = 13x + 6[/tex]

[tex]2x^2 + x = 13x + 6[/tex]

[tex]2x^2 - 12x - 6 = 0[/tex]

[tex]x^2 - 6x - 3 = 0[/tex]

By the quadratic formula,

[tex]x = 3 \pm 2\sqrt{3}[/tex]

but x must be a positive number, so only the positive square root solution is valid.

[tex]x = 3 + 2\sqrt{3}[/tex]

We identify a = c = 3 and b = 2, so a + b + c = 8.


Related Questions

help ! ✬

due tomorrow, i font understand this at all :(

Answers

Answer:

41 litters to gallons is 10.8311

3.5 gallons to litters is 13.2489

A = 10.8311 Gallons

B = 13.2489 Literes

A new cell phone costs $108.99 in the store. What would your total cost be if the sales tax is 4.5%? Round your answer to the nearest cent, if necessary.

Answers

Answer: $113.89

Step-by-step explanation:

108.99 * 0.045

= 4.90455

108.99 + 4.90455

= 113.89455

$113.89

Answer:

Total Cost = $113.89

Step-by-step explanation:

Total Cost = Cell phone cost + sale Tax % ( cell phone cost)

Total Cost = $108. 99 + .045 × $108.99

Total Cost = $108.99 + $4.90455

Total Cost = $108.99 + $4.90

Total Cost = $113.89

As an estimation we are told 5 miles is 8 km.
Convert 52.5 miles to km.

PLEASE HELP NO TIME FOR LINK!!!

Answers

Answer:

Step-by-step explanation:

52.5 mi(8 km/5mi) = 84 km

debbie used 40% of her flour for
cookies she started with 24 cups how many cups did debbie use?

Answers

Answer:

24 x 40% = 9.6

Step-by-step explanation:

the answer is Lil Debbie™ 9.6 cup used.

A car salesman earns a 3% commission on sales. If he sells a car for AED 75,990, how much the commission amount will he earn if he sells 4 cars?

Answers

Answer:

AED 9118.80

Step-by-step explanation:

to find out how much he earns for 1 car, multiply 75990 by 0.03 (3%). so the salesman earns 2279.70 per car sold. next, multiply this number by 4 to get the amount he earned for 4 cars: 9118.80

What is Two-thirds divided by one-sixth?
An area model with 4 shaded parts and 2 unshaded parts. The shaded parts are labeled two-thirds.
2
3
4
6

Answers

Answer: 4

Step-by-step explanation:

So, what I do to divide fractions is to turn them into reciprocals, which means changing it into a multiplication problem and turning the second fraction upside down. So, [tex]\frac{2}{3}[/tex] x [tex]\frac{6}{1}[/tex]. We multiply across, so 2 x 6 is 12, and 3 x 1 is 3, which gets us to [tex]\frac{12}{3}[/tex]. Since this is an improper fraction, we can simplify by dividing 12 by 3 to get 4. Hope this helps! :)

Answer:

4 (C) hope this helps!!

BYEEEEEEEEE!

I need help I can’t find the anwer please help

Answers

Answer:

You: 3, 6, 9, 12

Friend:4, 8, 12, 16

Step-by-step explanation:

You/Friend ratio is 3/4

Missing numbers:

8 / (4/3) = 612 * (4/3) = 16

Numbers:

You      3, 6, 9, 12

Friend  4, 8, 12, 16

Two teams are going on a trip to play a game. There will be
2 buses to take the students. On each bus, there are 60
seats. Teachers and parents will sit in 10 of the seats on
each bus
How many seats are available for students on the buses?

No links

Answers

-——————-
*%~.Answer .~%*
The Answer is 100
-——————-

First you would Multiply 2 and 60
You can do that by adding 60 two times.

-——————-

60. It should look like this. If you didn’t get
+ 60. 120 then your answers is incorrect, or
——— wrong. ☺️
120

-——————-

Now you multiply 10 x 2 because the teacher’s and parents will take up 10 is the seats on each bus!
To make is easier add 10 and 10 you should get 20!

-——————-

Now subtract 20 from 120.

120. You should get 100! If you didn’t get
- 20. 100 then you're answer is incorrect or
———. Wrong, Once again.
100

——-————————-

You're answer is, 100! There will be 100 seats leftover or available for the students on the busses!

~ pls Mark brainiest, or at least say thanks!

• TheTrueNezuko.

Extra Points and Brainliest Award if answered!!!!
Gerald is testing to see if a possible rational root is an actual rational root of a polynomial. He uses synthetic division to determine the value of the remainder.
Which theorem is he using to test the value?

A. Linear Factorization Theorem

B. The Fundamental Theorem of Algebra

C. The Factor Theorem

D. The Remainder Theorem

Answers

Answer:

its C.

Step-by-step explanation:

please mark me brainlists

Gerald used the factor theorem to test the value. Therefore, option C is the correct answer.

Given that, Gerald is testing to see if a possible rational root is an actual rational root of a polynomial.

What is Synthetic Division?

Synthetic division is a method used to perform the division operation on polynomials when the divisor is a linear factor. One of the advantages of using this method over the traditional long method is that the synthetic division allows one to calculate without writing variables while performing the polynomial division, which also makes it an easier method in comparison to the long division.

The factor theorem states that if f(x) is a polynomial of degree n greater than or equal to 1, and 'a' is any real number, then (x - a) is a factor of f(x) if f(a) = 0. In other words, we can say that (x - a) is a factor of f(x) if f(a) = 0.

Therefore, option C is the correct answer.

To learn more about the factor theorem visit:

https://brainly.com/question/26113096.

#SPJ2

Suppose that a risk-free investment will make three future payments of $250 in 1 year, $250 in 2 years, and $250 in 3 years. Instructions: Round your answers to 2 decimal places. a. If the Federal Reserve has set the risk-free interest rate at 4 percent, what is the proper current price of this investment? $ b. What is the price of this investment if the Federal Reserve raises the risk-free interest rate to 6 percent? $

Answers

Answer:

la théorie des plaques est

The total present value will be $693.75

The total present value will be $668.23

What is Future Value?

Future value is the value of an asset or investment as of some future date. In other words, the future value is the sum of money that an investment will be worth, assuming a certain rate of return, after a specific amount of time (interest rate).

First, i= 4/100 = 0.04

n= 1, 2, 3

Amount= $250

So, Present Value = 250/ (1 + 0.04) +  250/ (1 + 0.04)² +  250/ (1 + 0.04)³

=  250/ 1.04 + 250/ 1.04² + 250/ 1.04³

= 240. 38 + 231.13 + 222.24

= $693.75

So, The total present value will be $693.75

Second,  i= 6/100 = 0.06

n= 1, 2, 3

Amount= $250

So, Present Value = 250/ (1 + 0.06) +  250/ (1 + 0.06)² +  250/ (1 + 0.06)³

=  250/ 1.06 + 250/ 1.06² + 250/ 1.06³

= 235. 84 +  222.49 + 209.90

= $668.23

So, The total present value will be $668.23

Learn more about Future Value here:

https://brainly.com/question/14860893

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Ernie deposits $9,800 into a pension fund. The fund pays a simple interest rate of 7%. What will the balance be after four years?

Answers

Answer:

$12,845.80

Step-by-step explanation:

take the first number and go +7%. repeat four times

hope this helps :-)

Find c
[tex]a+a=10\\a+b=35\\b-z=5\\z+a+b+c= 10[/tex]

Answers

C=-50 a=5 b=30 z=25

25+5+30+(-50)=10

Answer= -50

a= 5

b= 30

z= 25

c= -50

Jacob saw a model of a building that will be built in his city. The model has a height of 40 centimeters and will be built to a scale of 1:50. What is the actual height of the building? (1 m = 100 cm)

Answers

Answer:

20 m

Step-by-step explanation:

1/50 = 40/x

Cross multiply.

1 × x = 50 × 40

x = 2000

2000 cm × (1 m)/(100 cm) = 20 m

Answer: 20 m

PLEASE ASAP HELPPPP PICK THEM ALL!!

Answers

Answer:

B

Step-by-step explanation:

Could I get help with this?

Answers

Answer:

you would pay $1000 because insurance would cover the 1500 for medical, and it would pay 5000 of the 6000 for damage to the car

Step-by-step explanation:

Ophira’s insurance company has notified her that her premiums will increase due to a poor insurance score and a recent claim she filed. The recent claim has increased her premiums by 41%, and her poor insurance score will cost her an additional $25 per month. Ophira currently pays $815 per year for insurance. How much will she pay next year?


$1149.15
$1672.15
$1449.00
$1449.15

Answers

Answer:

$1449.15

Step-by-step explanation:

currently paying 815

to pay next year: 41% increase + 25 extra per month

⇒ 141%·815 + 12·25

    = 1149.15 + 300

    = 1449.15

which is the first question in brainly

Answers

Answer:

this question

Step-by-step explanation:

Answer:

The first question was this: https://brainly.com/question/420 ur welcome :)

Step-by-step explanation:

Need some help with these 2 math questions, They are related so best to finish one then work on the bisect. Thank you.

Answers

Answer:

∠RQT = 23.5

∠PQR  = 133°

Step-by-step explanation:

Question 1

Step 1 - Add angles together:

As all angles on a straight line must add up to 180°:

3x - 5 + x + 1 = 180

Add like terms together

3x + x - 5 + 1

4x - 4 = 180

Step 2 - Add 4 to both sides of the equation:

4x - 4 + 4 = 180 + 4

4x = 184

Step 3 - Divide both sides by 4:

[tex]\frac{4x}{4} = \frac{184}{4}[/tex]

x = 46

∠RQS = x + 1

∠RQS = 46 + 1

∠RQS = 47

Given the ray QT bisects ∠RQS, the angles must be the same on either side so:

47 ÷ 2 = ∠RQT

∠RQT = 23.5

Question 2

As we know the value of x:

∠PQR = 3x - 5

Plug known values in:

3x - 5

3(46) - 5

138 - 5

∠PQR  = 133°

This can also be calculated by subtracting known angle ∠RQS from 180:

180 - 47 = 133.

Hope this helps!

PLEASE NO LINKS DONT WASTE YOUR TIME IM NOT GONNA CLICK. Use the graph of the polynomial function to find the factored form of the
related polynomial. Assume it has no constant factor.

Answers

Answer:

B.

Step-by-step explanation:

factored form actually contains the x intercepts right in it. Basically whatever sets the whole equation equal to zero.

On the graph you can see that it passes through both positive and negative 4. so (x+4)(x-4) is the answer because plugging in 4 for x will make it zero. Same for plugging in negative 4

Hope this helps!

What is rhe relationship between a centimeter and a meter?

Answers

Answer:

A meter is 100x bigger than a centimeter.

Step-by-step explanation:

1 meter = 100 centimeters

Answer:

1metre=100 centimeter

Step-by-step explanation:

that is the relationship between centimeter and metre

In a school of 1,000 students, 600 study physics, 300 study Calculus, and 100 study both. What is the probability that someone who does not study calculus studies physics? Give your answer as a decimal to one decimal place.

Answers

The probability that someone who does not study calculus studies physics is 0.5

Since there are 1000 students in the school and 600 study physics, 300 study Calculus, and 100 study both, we need to find the number of students that study only physics to find the probability that someone who does not study calculus studies physics.

Let x = number of students who study both = 100, y = number of students who studies only physics and z = number of students who studies only calculus.

Given that the number of students who study physics are 600, we have

x + y = 600  (1)

Also, given that the number of students who study calculus are 300, we have

x + z = 300  (2)

From (1)

y = 600 - x

y = 600 - 100

y = 500

So, the probability of students that study only physics = probability that someone who does not study calculus studies physics  is

P(physics) = number of students studying only physics/total number of students

= 500/1000

= 5/10

= 0.5

So, probability that someone who does not study calculus studies physics is 0.5

Learn more about probability here:

https://brainly.com/question/19916581

no link no bot please right fast but right

Answers

Answer:

12 in

Step-by-step explanation:

3in + 3in + 3in + 3in = 12

The perimeter is 12 inches because each side is 3 inches, therefore 3+3+3+3=12

Find the angles of a triangle which are in the ratio 4:3:2.​

Answers

Answer:

80o, 60o, 40o

Step-by-step explanation:

The sum has to add up to 180 degrees because of the three sides and this is what matches.

Answer:

Let the given angles of the triangle be 4x, 3x and 2x.

We know that the sum of all interior angles of a triangle is 180°.

Therefore,

4x+3x+2x = 1809x = 180x = 180/9x = 20

Hence, the angles of the triangle are -

4x = 4×20 = 80°3x = 3×20 = 60°2x = 2×20 = 40°

I’ll make you to Brainlius if you answer the question no links I’ll take a folder but no links or power point As long as you know the question and you didn’t the quiz at least send either PowerPoint or a folder please I need help immediately ASAP

Answers

Answer:

1) Is 9 2)x-3   3) 12 ,106,42 4)1/12 5) one ,three half ,three 6)32 7) 1,4,5 8) 1 z-9 2 9+z 3 9/3 4 9z 9) 3/n 10 10) 1 2

Step-by-step explanation: Cuz I'm In The Test Now And Jest Got Done With It And All The Answers are Right here  

2^3+4•2-7=
2^4=8
4•2=8
8+8=16-7
16-7=9

The perimeter of a rectangle is 21+ 2where is the
length of the rectangle and wis the width find the
Perimeter of the rectangle if the length is 12 inches and the width is 9 inches

Answers

Answer:19 - 20

Step-by-step explanation:

Yo

Fill in the missing
numbers?
?•(-10) = 90

Answers

Answer:

?=-9

Step-by-step explanation:

A negative x negative is positive. 90/10=9 and mutiply both by -1. -10x-9=-90

how many times does 32 go into 7500​

Answers

Answer:

yy

Step-by-step explanation:

Answer:

234.375

Step-by-step explanation:

divide 7500 by 32, and you'll get the number of times 32 goes into 7500.

7500 ÷ 32 = 234.375

and if the question is asking for whole numbers it would be just 234 times (and a 3/8)

On Saturday, 150 people at a movie theater purchased popcorn, and 350 people did not purchase popcorn. What percentage of people did not order popcorn at the movie theater on Saturday?​

Answers

Answer:

70% of people did not order popcorn at the movie theater on Saturday.

Step-by-step explanation:

first, add to get the total amount of people

150 + 350 = 500

now that we have the total, divide the people that did not buy popcorn by the total amount of people

350/500 = 0.7

then, multiply that by 100

0.7 x 100 = 70

your final answer would be 70%

Please Help!
what is the resultant of the two vectors shown above?​

A. (9,1)
B. (1,-9)
C. (-1,9)
D. (5,5)

Answers

Answer:

D 5 5 it has to be the answer

The diameter of Pilar's bike wheel is 24 inches.
How far does Pilar's bike travel when the wheel makes
10 complete rotations?

Answers

The distance Pilar's bike travel when the wheel makes  10 complete rotations is 753.6 inches

Since the wheel is circular in nature, we will take the circumference to determine the distance travelled for 1 revolution

Circumference of a circle = πr²

r is the radius of the wheel.

Given that d = 24 in, r = 24/2 = 12in

C = 2(3.14)(12)

C = 75.36 in

Hence the distance completed by 1 revolution is 75.36. If Pilar's bike makes 10 revolutions, hence the required distance will be:

D = 10 * 75.36

D = 753.6in

Hence the distance Pilar's bike travel when the wheel makes  10 complete rotations is 753.6 inches

Learn more here: https://brainly.com/question/23242098

Other Questions
Read the passage from a story in The Arabian Nights Entertainments.The fisherman was very unhappy. "What an unlucky man I am to have freed you! I implore you to spare my life.""I have told you," said the genius, "that it is impossible. Choose quickly; you are wasting time."The fisherman began to devise a plot."Since I must die," he said, "before I choose the manner of my death, I conjure you on your honour to tell me if you really were in that vase?""Yes, I was," answered the genius."I really cannot believe it," said the fisherman. "That vase could not contain one of your feet even, and how could your whole body go in? I cannot believe it unless I see you do the thing."Then the genius began to change himself into smoke, which, as before, spread over the sea and the shore, and which, then collecting itself together, began to go back into the vase slowly and evenly till there was nothing left outside. Then a voice came from the vase which said to the fisherman, "Well, unbelieving fisherman, here I am in the vase; do you believe me now?"What is the fishermans motivation for asking the genius if he really was in the vase?to make the genius grant him three wishesto try to find a way to stay aliveto see if the genius was lyingto have more time before the genius kills himMark this and return What should a bank statement include what gave Lincoln the authority to excuse the Confederates of treason Help! This assignment is due at 7:00 and this is the last question! Solve for all values of b in simplest form (delta math)|6-3b|=4 In Amigo Brothers, what does "Let the best man win" mean? HELP ASAPSolve for the measure of PMR. Show your work and explain how you got your answer. A hardware store owner placed an advertisement for Sylvania LED bulbs in the local newspaper. Sylvania provided the storeowner with several high-quality, camera-ready sample advertisements. In addition to supplying the advertising formats, Sylvania also paid 50 percent of the cost to place the ad in the paper. Sylvania was using ________ to promote its products. manufacturer incentive advertising strategic channel partnership reseller consumer promotion retail publicity cooperative advertising Can you help me please I will mark the BRAINLIEST At Arnold's Arcade, a child can buy 21 tokens for $6. At this rate, how many tokens would $20 get someone? what do you mean by blood pressure is 130 19 mm of HG People in the New England colonies mostly farmed because of the fertile farmland. O True False a.Which sentence below has no comma errors?He seems, to be such a lonely quiet man, doesn't he?b. To apply, for this job you must have a social security card.C. You said I could go, didn't you?Please select the best answer from the choices provided Bonnie read an informational text about the cello for music class. In her analysis, she made the following claim:The cello is a difficult instrument to play.Which explicit evidence from the text would best support Bonnies analysis?A) A cellist must learn accurate finger placement by ear and practice for several hours daily.B) The cello has four strings that lie across the bridge piece; it does not have marked frets.C) The cello is a string instrument, like a violin, and plays mostly on the bass clef of the music staff.D) YoYo Ma is a famous modern cellist, and his collaborations with popular artists have helped revive interest in the cello. Diana exhibits two separate personalities, each well-defined with distinct characteristics. She could be diagnosed with ________ disorder. help extremely quick plsssssssssssssssssssssssssssssssssssSelect all the expressions that have a value of 180 when b = 4.A. 87b - 179 + 957 87B. 33b/12C. b + 120D. 51b - 24E. 11b + 187 - 2,756 13 although a fetus is relatively protected in the mothers womb a number of different toxins can harm the fetus What is the line that goes through (2,3) and is perpendicular to the line y = -2x + 9? ASAP :) Which sorts of countries tend to benefit from a federal system?. Which method of bond amortization amortizes the premiums/discounts accurately and is considered a conceptually superior method Which of the following metabolic processes uses the most energy?A. Growth and repairB. DigestionC. Exercise and warmthD. Excretion