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

Solve the following of:

11/15 × 7/8 ÷ 3/4​

Answers

Answer:

Well following the order of operations

You just do this from left to right

11/15* 7/8 = 77/120

77/120 ÷ 3/4

or

77/120 * 4/3

77/90 (simplified)

77/90 is the answer

Match to the other shape​

Answers

Answer:

wheres the picture?

Step-by-step explanation:

How to determine the domain and range of a continuous graph

Answers

Answer:

Step-by-step explanation:

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis.                                        

On a coordinate grid, point L is at (−4, 4) and point S is at (8, 4). The distance (in units) between points L and S is ______. Input only whole numbers, such as 2.

Answers

Answer:

12 this is da a answer

12 twelve

12x3y2z3 - 8x2y2z2 + 20xy2z3

Answers

The answer is 24x+2y+2z+4.

The graph of a linear is shown below on the coordinate grid. What is the y-intercept of the graph of the linear equation? Bubble is your answer.​

Answers

Answer: -7 on the y-axis

Step-by-step explanation:

how do I solve this?

Answers

Its g is increasing

I guess you could just plug in 0, 1 and -1 and graph it.

paulita reads an average of 20 pages each day she has 6 days to read 10 pages will she finish her reading in 6 days explain?

Answers

Answer:

Yes

Step-by-step explanation:

If she is reading 20 pages a day, and needs to read 10 pages, then she needs half a day, and since she has 6 days, she can do this.

A corporation recently sold zero-coupon $400 bonds maturing in 4 years with an annual yield of 12%. Find the price. [Hint: The price is the present value of $400 4 years from now at the stated interest rate.] (Round your answer to the nearest cent.)

Answers

Answer:

Step-by-step explanation:

400*1.12^-4= 254.2072314

help me ! ✬

work due tomorrow. i don’t understand this at all.

there’s 2 questions !


— don’t comment for points.

Answers

Answer:

32, 40

Step-by-step explanation:

See the image

Hope that helps! :)

Please find the value of each of the variables. Show all work.

Answers

Answer:

W=20 V=42 X=87

Step-by-step explanation:

all sides are equal meaning that two corners from across each other are equal so v is 42 and w 20 this also means x is 87 im not too sure about y

I need to find out the answer for this problem...

Answers

Answer:

where is the problem to solve. please tell me

string is 38 cm longer than his desk. Length of his reading book is 26 cm, what is the length of his string?

Answers

Answer:

has in lenght of 2 or 5

Step-by-step explanation:

odyssey ware

ig i d k

solve the equation 5/x+3 + 4/x+2 = 2

Answers

Answer:

5 + 4 = 9

x × x = X2

3 + 2 = 5

9/x + 5 = 2

Answer:

5/x+3+4/x+2=2

=9/x+5=2

9/x=-3

Multiply both sides by x

9= -3x

Divide by -3

Answer x=  -3

Hope this helps

-7+0.3(-4x+20)+8.8x
Help me plz
This is 8th grade math btw

Answers

Answer:

7.6x−1

Step-by-step explanation:

Expand by distributing terms

−7−1.2x+6+8.8x

Collect like terms.

(−7+6)+(−1.2x+8.8x)

Simplify

−1+7.6x

Regroup terms.

7.6x−1

sarah had $40 at the beginning of the week. She babysat 3 days and by Saturday had $110. what is the percent increase

Answers

Answer:

175% increase

Step-by-step explanation:

How many solutions −2(+1)=2+5

Answers

Answer:

no solutions

Step-by-step explanation:

-2+1=2+5

-1=7

-1 does not equal 7 so there are no solutions

Qiang is inviting 4 friends to a party each friend wants 4 cookies each box has 10 cookies how many boxes should qiang get

Answers

2
4 friends and each wants 4 so total cookies
required is 16. 1 box has 10 cooking which is not enough,
so you need to get 2 boxes to have more than
16 cookies

what is 10 divided by 8 equal?​

Answers

Answer:

it is equal to 1.25

Step-by-step explanation:

hope it's helpful

Answer:

Decimal:

1.25

Mix Number:

1 1/4

Improper Fraction:

5/4

Step-by-step explanation:

10/8 is 1.25

In fraction it is  1 1/4 or 5/4

Hope this helps!

Eric ran 1 1/4 miles. Then he biked 3 1/2 times farther than he ran.


How far did Eric bike?


Enter your answer as a mixed number in simplest form.

Answers

Answer: He biked 4 and 3/8 miles

Step-by-step explanation:

since he ran 1.25 miles and biked 3.5 times farther than that you'd multiply both numbers and get 4.375 which simplifies to 4 and 3/8

How do I do this question!

Answers

Answer:2 x 3

Step-by-step explanation:

Answer: [tex]\frac{7}{10}[/tex]

Step-by-step explanation:

2[tex]\frac{1}{10}[/tex] barrels of almonds to make 3 batches of granola bar.

We need to turn fraction 2[tex]\frac{1}{10}[/tex] into a mixed number

2[tex]\frac{1}{10}[/tex] → [tex]\frac{21}{10}[/tex]

Now we set up an equation.

[tex]\frac{\frac{21}{10} }{3} =\frac{x}{1}[/tex]

Solve for x by cross multiplying and dividing.

[tex]\frac{21}{10}[/tex] ÷ 3 (KCF)

[tex]\frac{21}{10}[/tex] × [tex]\frac{1}{3}[/tex] = [tex]\frac{21}{30}[/tex]

Now we simplify the fraction.

[tex]\frac{21}{30}[/tex] → [tex]\frac{7}{10}[/tex]

[tex]\frac{7}{10}[/tex] barrels of almonds were used in each batch.

If the smaller of two numbers is one-half of the larger number and the sum of the two numbers is 78, find the numbers

Answers

divide 78 by 3
= 26
the numbers are 26 and 52

Answer:

52 and 26

Step-by-step explanation:

2x=y

x+y=78

3x=78

x=26

2(26)=y

y=52

To check:

52+26=78

Graph algebra problem! Can anybody help me out as graphs are definitely not my strong suit.​

Answers

Answer:

f(2) = 5

Step-by-step explanation:

Definitions:

Input values = x-coordinates.

The input values are also known as the domain, which is the set of all real numbers. With the domain of a function, you could substitute any real number into a given function for x and produce a valid output.

Output values = y-coordinates

The range is the set of all real numbers that depend on the input values plugged into the function. Thus, the range represents the output (y-values) that corresponds to the input values used into the function. Function Notation:In the function notation f(x), f  is the function name, and x is your input variable.  Output values are also called functional values.  You could use any letter to represent a function name, such as g(x), w(x), etc.,.Explanations:

This particular question asks for you to identify the corresponding output (y-value) of the given input value, x = 2.  If you look at the graph, the x-coordinate, x = 2, corresponds to the y-coordinate of 5.

Therefore, f(2) = 5.  

Attached is an edited screenshot of the graph that you posted, where it shows where f(2) = 5 is located on the graph.

Answer:

  f(2) = 5

Step-by-step explanation:

The 2 in f(2) means you're looking for the value of the function when x=2.

Find the vertical line that is the 2nd one to the right of the y-axis. Follow it up until you reach the function line (at the top edge of the graph). The y-value there is the value of f(2). That is 5 squares above the x-axis, so ...

  f(2) = 5

The half-life of radium is 1690 years. If 80 grams are present now, how much will be present in 870 years?
grams

Answers

Answer:

A(t) = amount remaining in t years

= A0ekt, where A0 is the initial amount and k is a constant to be determined.

Since A(1690) = (1/2)A0 and A0 = 80,

we have 40 = 80e1690k

1/2 = e1690k

ln(1/2) = 1690k

k = -0.0004

So, A(t) = 80e-0.0004t

Therefore, A(430) = 80e-0.0004(430)

= 80e-0.172

≈ 67.4 g

Step-by-step explanation:

hope it helpful for you ❤️

Teresa piano recital starts at 2:41 p.m. It ended at 6:36 pm how long did the recital last

Answers

Answer:

way too long

Step-by-step explanation:

jk, 3 hours and 55 minutes

Which transformations map the strip pattern onto itself?
PPPPP
O
a horizontal translation only
a horizontal translation and a glide reflection
a horizontal translation and a reflection across a vertical line
a horizontal translation and a 180° rotation

Answers

Answer:

2nd is correct

But u can also write by guessing. it is the best trick used by toppers.

Can someone help me please

Answers

Answer:

52.7%

Step-by-step explanation:

If I am correct it means to round the percentage, so it would be 58% if you are rounding it

Hey, wassup its sofie dossi! I would like to ask you what 567865676545678+456789098765432347898789 is?

Answers

Answer:

828383627262827294688

Step-by-step explanation:

When you add 567865676545678 to 456789098765432347898789, the best way to do it is by placing the biggest number above the smallest number on a paper. Visual representation does help!

To be clear, I am meaning in a format such as this;

456789098765432347898789

+                  567865676545678

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

Then, just as you would add any simple 2 number equation such as 3+5, etc. you will look at these numbers from right to left. Look at the bottom right one first, then add it to the one above it, if it adds up above 9, you will carry the leftover number above the row to the left of your current row. So currently you are in row 1, and you will add up to get 17. That means you will keep the 7 and write it down at the bottom and carry the one to the next row, and place it above the 8.

Hope this helps!

expression:
x + 20 + x + 20 + x + 20 + x + 20

Answers

Answer: if is simplify the answer is 4x+80

Step-by-step explanation:

A rectangular board is 1100 millimeters long and 800 millimeters wide. What is the area of the board in square meters? Do not round

Answers

Answer:

880 square meters

Step-by-step explanation:

You need to find the area of the rectangular board, which would be 1100x800=880000 square millimeters. 1 meter=1000 millimeters, so you divide 880000 by 1000, getting 880 square meters.

The area of the board is 880 square meters.

Area of rectangle

You need to find the area of the rectangular board, which would be 1100x800=880000 square millimeters.

1 meter=1000 millimeters, so you divide 880000 by 1000, getting 880 square meters.

Learn more about area of rectangle here https://brainly.com/question/25292087

#SPJ2

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