[tex]\sf\:Q.\:Show \: that \: there \: is \: no \: positive\:integer[/tex]
[tex]\sf\:n \: for \: which \:\sqrt{n - 1} \: + \:\sqrt{n + 1}\:is[/tex]
[tex]\sf\: rational.[/tex]​

Answers

Answer 1

9514 1404 393

Explanation:

We will prove by contradiction. We assume that the given sum is rational, and the ratio can be expressed in reduced form by p/q, where p and q have no common factors.

  [tex]\sqrt{n-1}+\sqrt{n+1}=\dfrac{p}{q}\qquad\text{given}\\\\(n-1)+2\sqrt{(n-1)(n+1)}+(n+1)=\dfrac{p^2}{q^2}\quad\text{square both sides}\\\\2(n+\sqrt{n^2-1})=\dfrac{p^2}{q^2}\qquad\text{simplify}[/tex]

We note that this last equation can have no integer solutions (n, p, q) for a couple of reasons:

for any integer n > 1, the root √(n²-1) is irrational (n² is a perfect square; n²-1 cannot be.)p²/q² cannot have a factor of 2, as √2 is irrational

There can be no integer n for which the given expression is rational.


Related Questions

Which point of concurrency is NOT always inside a triangle? Check all that apply. A. circumcenter B. incenter C. orthocenter D. centroid

Answers

The point of concurrency which is not always inside a triangle is circumcenter and orthocenter.

We are given that;

The four options

Now,

A point of concurrency is a single point shared by three or more lines1. In a triangle, there are four points of concurrency that are associated with different types of line segments: centroid, incenter, circumcenter and orthocenter.

The centroid is the point of concurrency of the medians of a triangle, which are the line segments that connect the midpoints of the sides to the opposite vertices. The centroid is always inside the triangle.

The incenter is the point of concurrency of the angle bisectors of the triangle, which are the line segments that divide the angles into two equal parts. The incenter is also always inside the triangle.

The circumcenter is the point of concurrency of the perpendicular bisectors of the sides of the triangle, which are the line segments that are perpendicular to the sides and pass through their midpoints.

The circumcenter can be inside or outside the triangle, depending on the type of triangle. For an acute triangle, it is inside; for a right triangle, it is at the midpoint of the hypotenuse; for an obtuse triangle, it is outside

Therefore, by triangle the answer will be circumcenter and orthocenter.

Learn more about the triangle;

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The housing sales measured in millions of dollars, M, of a rural city, can be modeled by the function M(t) = -35cos(0.524t) + 55, where t represents the number of years since 2020. What characteristics of M(t) are true?
I. The first time the housing sales will reach their maximum will be in 2030.
II. Housing sales are expected to increase from 2020 to 2021.
III. The range is (20,90).

Answers

Answer:

It is going to be the first one.

Step-by-step explanation:

The first time the housing sales will reach their maximum will be in 2030.

Hope it helps! :)

Hope you also have a great rest of your day!

yuurrrr i need help yo plz help

Answers

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Answer: CD = 90

Explanation:

4b-30=3b

b = 30

3b = CD

3(30) = 90

CD = 90

I hope this helped!

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Answer:

90

Step-by-step explanation:

it is a square and all the edges have same len;

4b-30 = 3b , b = 30

.

3b = 90

Would someone be able to answer this before 3 PM central?

Answers

Answer:

yes

Step-by-step explanation:

7. Alex has 4 red marbles 3 blue marbles, 2 yellow marbles, 6 purple marbles, and one white marble. Alex picked a marble out of the bag, recorded the color in the chart (to the right), then replaced the marble. He repeated this experiment fifteen times. COLORS FREQUENCY
What is Alex's experimental probability of Red getting a blue?

b. What is Alex's theoretical probability of getting a blue?​

Answers

Answer:

B

Step-by-step explanation:

4. Vince is buying apples for his horse. Which of these options has the
lowest unit cost?
A. 5 apples for $2.50
c. 12 apples for $7.20
B. 8 apples for $4.40
D. 1 apple for $0.52
(Z.RP.A.1
usina
Alions of as How

Answers

Answer:

option A. 5 apples for $2.50

Step-by-step explanation:

You must find the price per apple for each option by dividing the cost in cents by the number of apples. 250 cents divided by 5 apples gives you 50 cents per apple, while option c is 55 cents per apple, option b is 60 cents per apple, and option d is 52 cents per apple.

if a rectangular square pyramid has the square base of 3in and the height of 4 in, what is the volume?

Answers

Volume of a pyramid = Length x width x height / 3

A square base means both the length and width are the same.

Volume = 3 x 3 x 4 /3 = 12 square inches.

Answer:

12in^3

Step-by-step explanation:

v = (1/3) b^2 * H

b^2 = base length^2

(1/3) 3^2 * 4 = 12

1/3 of 9 = 3 x 4 = 12

HOW DO YOU CONVERT SURFACE AREA TO VOLUME!
HELP!!!!
WILL MARK BRAINLIST!!

Answers

Answer:

Formulas for a rectangular prism:

Volume of Rectangular Prism: V = lwh.

Surface Area of Rectangular Prism: S = 2(lw + lh + wh)

Space Diagonal of Rectangular Prism: (similar to the distance between 2 points) d = √(l2 + w2 + h2)

Step-by-step explanation:

Hope it helps :))))

help me plz !!!
what´s 6×6×6×6=

Answers

Answer:

hey child !! you should practice some maths okay.

6 × 6 × 6 × 6 = 1296

24

6 × 6 × 6 × 6 = 24

hope it helps

A system of three linear equations in three variables is consistent and independent. How many solutions to the system
exist?

A.None
B.one
C.three
D.infinitely many

Answers

B........................

Plz help with this problem

Answers

Answer:

33 % is the answer.

Step-by-step explanation:

The value provided is 10 and possible out comes of getting a result is 3 as there are 3 grades namely (A,B and C) are mentioned. When converted to 100s, 100/33 gives 33.33% as the answer, when rounding off gives 33.

HELP PLEASE ASAP THANKS

Answers

Answer:

188.4

Step-by-step explanation:

(4x8) + 6


Easy points ✨✨✨

Answers

Answer:

38

Step-by-step explanation:

You need to follow the rule of bodmas.

Brackets first, then multiplication and division, then addition and subtraction.

(4 x 8) + 6 = 32 + 6 = 38

Answer:

38

Step-by-step explanation:

(4 x 8) + 6

32 + 6

38

Can someone please answer quick. I only have two minutes to answer the question.

Answers

it’s -4. because it passes through the vertex. it’s x=-4

Find x. Round your answer to the nearest tenth of a degree and show work.

Answers

Answer:

35.0

Step-by-step explanation:

14+21 equals 35 and then round to the nearest tenth of a degree and came up with 35.0.

What is the margin of error for a sample statistic for a population with a standard deviation of 6.75?

Answers

Answer:

Step-by-Margin of Error  = 1.96 x  7.75 = 15.19

step explanation:

i) Margin of Error of a population with a standard deviation of 7.75?

ii) Let us assume a two tailed test with a 95% confidence interval

iii)  

iv) therefore Margin of Error  = 1.96  7.75 = 15.19

Answer: 13.5

Step-by-step explanation:

X
Combine like terms:
b + 4b2 - 8 + 3b - 2b2

Answers

Answer:

[tex]2b^{2}[/tex] + 4b - 8

Find the distance between (-1,-2) and (8,9)

Answers

Answer:

[tex]\sqrt{202}[/tex] units or 14.2 units (rounded to the nearest tenth)

Step-by-step explanation:

To find the distance between a pair of two points, use the distance formula  [tex]d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]. Substitute the x and y values of (-1,-2) and (8,9) into the formula and simplify:

[tex]d = \sqrt{(-1-8)^2+(-2-9)^2} \\d = \sqrt{(-9)^2+(-11)^2} \\d = \sqrt{81+121} \\d= \sqrt{202}[/tex]  

So, as an exact answer, the distance is [tex]\sqrt{202}[/tex] units.

(To find the decimal approximate of that answer, enter it into a calculator. This would make the decimal approximate 14.2 units, when rounded to the nearest tenth.)  

Answer:

14.2

Step-by-step explanation:

All you need to do is plug the coordinates into the distance formula.

√(x2 - x1)² + (y2 - y1)²

√[8 - (-1)]² + [9 - (-2)]²

√(9)² + (11)²

√81 + 121

√202

= 14.2

HELPPPPPP this is due tonight!!!!

Answers

Answer:

(1) Commutative property

(2) Distributive property/Definition of addition

(3) Compatibility with addition/Existence of additive inverse/Modulative property

(4) Compatibility with multiplication/Associative property/Definition of division/Existence of additive inverse/Commutative and modulative properties/Result

Step-by-step explanation:

We proceed to solve algebraically and explain each step:

1) [tex]4\cdot x +1 +x + 3 = -11[/tex] GIven

2) [tex]4\cdot x +x +1 + 3 = -11[/tex] Commutative property

3) [tex](4+1)\cdot x +4 = -11[/tex] Distributive property/Definition of addition

4) [tex]5\cdot x +4 = -11[/tex] Definition of addition

5) [tex]5\cdot x +4 -4 = -11-4[/tex] Compatibility with addition

6) [tex]5\cdot x = -15[/tex] Existence of additive inverse/Modulative property

7) [tex]\frac{1}{5}\cdot (5\cdot x)= \frac{1}{5}\cdot (-15)[/tex] Compatibility with multiplication

8) [tex]x = - 3[/tex] Associative property/Definition of division/Existence of additive inverse/Commutative and modulative properties/Result

Hence, we have the following answers:

(1) Commutative property

(2) Distributive property/Definition of addition

(3) Compatibility with addition/Existence of additive inverse/Modulative property

(4) Compatibility with multiplication/Associative property/Definition of division/Existence of additive inverse/Commutative and modulative properties/Result

Jason borrows $8,000 to help pay for college. If the loan has an annual interest rate of 5.5%, how much interest would he owe at the end of two years, if he does not make any payments?

Answers

Answer:

the interest that would he owe at the end of the two years is $880

Step-by-step explanation:

The computation of the interest that would be owe at the end of two years is shown below

here we applied the simple interest formula

= Principal × rate of interest × time period

= $8,000 × 5.5% × 2

= $880

Hence, the interest that would he owe at the end of the two years is $880

Please help me explain me this plsss!!!!

Answers

The anser is C for reasons of science ;)


Help me pleasseee!!!

Answers

Answer:

Using x=5, then x exponent 2 = 5 exponent2 =5 x5 = 25 1 exponent5=1x1x1x1x1=1 5 exponent 1 = 5 my sister said the answer is 162 you could try that

Answer: 25, 1, 5

Explanation: you change all the x's to 5
*for x^2 you get 5^2 which is just 5x5 so it's 25.
*for 1^x you get 1^5 which is just 1x1x1x1x1 so it's 1
*for x^1 you get 5^1 which is just 5

Hope this helped!

PLEASE GOD HELP ME THE FIRST GOOD ANSWER WILL BE MARKED BRAINIEST

Answers

Answer:

m∠BAC = 52°  or theta = 52°

Step-by-step explanation:

I am guessing that you are trying to find m∠BAC ( which is theta)

(1)  The measure of an inscribed angle is 1/2 the measure of the intercepted arc

measure of arc BC is twice the measure of ∠CDB

measure of arc BC = 2(26) = 52

(2)  The measure of a central angle equals the measure of the intercepted arc.

m∠BAC = 52°

Note:  You need to remember statements (1) and (2) to help you with this type of problem.  OK?

Write an explicit formula for the following sequences.
-4, -6, -8, -10...

Answers

Answer:

tn=-2n-2

hope this helps :)

Plz help me with this

Answers

9514 1404 393

Answer:

  d)  -4 ≤ x ≤ 1.5

Step-by-step explanation:

The domain is the horizontal extent of the graph. Here, it goes from about x=-4 to about x = 1.5.

There are solid dots on both ends of the line segment, so the inequality will use ≤ for both limits. The domain is ...

  -4 ≤ x ≤ 1.5

please help i will mark branliest

Answers

Graph D look at the number line and see where 3.7 is

using long division: x^4+7x^3+13x^2+6x-45 / x+5

Can someone please help
Please show work

Answers

Answer:

x^3+2x^2+3x-9

Step-by-step explanation:

Equation practice with vertical angles solve for x

Answers

Answer:

x = -11

Step-by-step explanation:

6x+185 = 8x+207

substract 8 from both side and you will get

-2x = 22

and then divide by -2 and x will be equal to

x = -11

Draw the graph of the equation: 1.5x+2y=3

Answers

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Answer: See graph below!

I hope this helped!

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The graph of the given equation is attached to the solution.

What is graph?

In math, a graph can be defined as a pictorial representation or a diagram that represents data or values in an organized manner.

Given is an equation, 1.5x+2y = 3 we need to graph it,

So, first finding the coordinates,

1.5x+2y = 3,

Put x = 0

y = 1.5 [0, 1.5]

Put y = 0

x = 2 [2, 0]

Putting the coordinates on the graph and connect them, the line obtained will be the graph of the equation,

Since, the equation is linear the graph will be a straight line,

Learn more about graphs, click;

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Please help thank you

Answers

Answer B / 77. You need to look at where the line in the rectangle is protruding. Its marked that the line is aiming towards 77 on the graph.
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