Prove that (n+1)^2 - (n-1)^2 is always odd for all positive integer values of n.

Answers

Answer 1

Let's try it this way.
Do you know how to expand a binomial squared? If you do here's one way to do it.
(n +1)^2 = n^2 + 2n + 1
(n - 1)^2 = n^2 - 2n + 1

Now put them together
(n^2 + 2n +1) - (n^2 - 2n + 1) Remove the brackets.
(n^2 + 2n + 1 - n^2 + 2n - 1
2n + 2n = 4n is what is left. What you have given us is always even.

To show (but not to prove) that the result is even, let n = 10
(10 +1)^2 - (10 -1 )^2
(11)^2 - (9^2)
121 - 81 = 40

Maybe if you start out with (say) n = 7 (an odd number) you might get an odd number.
(7 + 1)^2 - (7 - 1)^2
8^2 - 6^2 (doesn't matter what the answer is. It will be even.
64 - 36 = 28 and that is also even.

Nice question. Maybe because you are in middle school mathematics, this last proof is good enough. In any case, you will never get an odd result. Isn't that something? Thanks for posting.

Related Questions

Select all of the angles that have the same measure as angle 1. Assume the lines are parallel.

Answers

Angle 3, 6, and 8 have the same measure as angle 1

Gracie has 1/2 of a cantaloupe that she wants to divide. How many 1/12’s of a cantaloupe are there in 1/2 of a cantaloupe?

Answers

there are 6
Divide bottoms or denominators
12 and 2
and you can get 6

there are 6 halfs because you have to divide denominator

A scale drawing of a rose is 3 inches long. The actual rose is 1.5 feet long. What is the scale factor of the drawing and what is the scale of the drawing? Please explain!! :D

Answers

so 1.5x=3
so, 3/1.5 is 2. since the rose (preimage) is smaller then the image, it is 2 because 1.5×2 is 3.

Answer:

The scale of the drawing is 6

Step-by-step explanation:

Given

Actual Length of Rose = 1.5 feet

Scale drawing of rose = 3 inches

First, we need to equalize both units (by converting feet to inches or inches to feet)

Convert the actual length of rose to inches

If 1 foot = 12 inches

Then 1.5 feet = 12 * 1.5 inches

1.5 feet = 18 inches

Now that we have a single unit for both measurement;

Using the scale factor formula

y = Cx,

where C is the scale factor for x.

y is the measurement of the actual rose = 18 inches

x is the scale drawing of the rose = 3 inches

By substitution; y = Cx, becomes

18 inches = C * 3 inches --- Make C the subject of formula

C = 18 inches / 3 inches

C = 6

Hence, the scale of the drawing is 6

I need to know the surface area of this shape

Answers

The area is the area of 2 pairs of congruent triangles and one rectangle. Use the formulas for the areas of these figures, then add results.

Front triangle area (A = (1/2)bh)
.. = (1/2)(16 m)*(25 m) = 200 m^2

Right side triangle
.. = (1/2)(22 m)*(24 m) = 264 m^2

Base (A = lw)
.. = (16 m)*(22 m) = 352 m^2

Total area
.. 2*front triangle +2*side triangle +base
.. = 2*200 m^2 +2*264 m^2 +352 m^2
.. = 1280 m^2

use the distributive property to rewrite 30d+18

Answers

Im assuming you just want to take out common multiples so,
30d+18 is also the same as 6(5d+3)

what is the equation to (-3,1) and (0,-8)

Answers

The change in x between the two is 3 and the change in y is -9.
Divide -9 by 3 to get -3 using the rise over run slope formula.
The equation is y = -8 + -3x because the -8 in the second set of coordinates is the y-intercept of the equation.
Hope this helps!

The ratio of youths to adults at a skating rink is 13:7. What percent of these people are youths?

Answers

The percentage of people at the scathing ring that are Youth is 65%. You would have to add 13 + 7, to get 20. Then you do 100 percent divided by 20 to get 5. Finally, you would do 5 times 13 to reach 65%.

65%, is the answer why is the question/??? BECAUSE I SAID SO

how do you solve 6) n2+7n+15=5 by factoring?

Answers

First, you need to set the equation equal to zero:

n^2 + 7n + 10 = 0

Now we factor. We need to find two numbers that add up to 7 and multiply to 10.

2 + 5 = 7
2 * 5 = 10

Now, we just need to write this as a polynomial:

(n + 2) (n + 5)
is our answer.

Hope this helps!

plzzzzzzzzzzzzzzz help

Answers

One
===
You need to get the total to 6/8
Cats = 1/8
Dogs = 5/8 The total here is 6/8

Two
===
Now you need the total to go to 5/8
4/8 + 1/8 will do it.
Cats = 4/8
Dogs = 1/8

a rectangular prism has a volume of 300 cm squared if the scale factor of the second rectangular prism is 3.5 what is the volume of the second rectangular prism

Answers

Volume of rectangular prism= 30 cm³
Formula for volume of rectangular prism:
V=whl
Scale factor of second prism= 3.5
Which is 3 and a half times greater than the 1st one.
So volume of 2nd prism = 3. 5 x 300
V= 1050 cm³

Answer: Volume of second Rectangular Prism is 1050 cm³

(06.06) Which of the following demonstrates the Distributive Property? (4 points) 3(4a + 2) = 12a + 6 3(4a + 2) = 12a + 2 3(4a + 2) = 4a + 6 3(4a + 2) = 7a + 5

Answers

3(4a+2)=12a+6 shows the distributive property.

Answer:

could u be more spasific?

Step-by-step explanation:

Olga wrote all the natural numbers from 1 to k. Including 1 and k. How many numbers did she write?

Answers

Given that "Olga wrote all the natural numbers from 1 to k. Including 1 and k"

So Olga write k numbers.

lets take an example

if k = 10 , then natural numbers between 1 and 10 are 10 numbers that are 1 , 2 , 3 ,4 ,5 ,6 , 7 , 8 ,9 and 10.

Hence she write k numbers.

Solve for θ if sinθ = 0.45. Round answer to the nearest degree.

Answers

If you look in a table of sines, you find that sin(θ) = 0.45 corresponds to an angle of about 26°45', so is nearest to 27°.

7x+39≥53 AND 16x+15>31

Answers

Answer: x \> 2

Just look in the photo

I'll give brainliest if you show your answer. Plz answer before thirty minutes.
The means and mean absolute deviations of the individual times of members of two relay swim teams are shown in the table below.
Means and Mean Absolute Deviations of
Individual Times of Members of 4*200- meter Relay Swim Teams

Team A Team B
Mean 127.9 s 127.4 s
Mean Absolute Deviation 0.26 s 0.23 s
The difference of the means is found and then compared to each of the mean absolute deviations. Which is true?
The difference between the mean times is about equal to the mean absolute deviation of the data sets.
The difference between the mean times is about 2 times the mean absolute deviation of the data sets.
The difference between the mean times is about 5 times the mean absolute deviation of the data sets.
The difference between the mean times is about 16 times the mean absolute deviation of the data sets.

Answers

The difference between the mean times is about 16 times the mean absolute deviation of the data set.

127.9-127.4=0.5
0.26-0.23=0.03
0.5/0.03=16.6666

About 16 times

Answer:

The difference between the mean times is about 16 times the mean absolute deviation of the data sets.

Step-by-step explanation:

We are given the mean and mean absolute deviation of two teams in a table as:

Team A Team B

Mean 127.9 s 127.4 s

Mean Absolute Deviation 0.26 s 0.23 s

The difference of mean between two team is: 0.5 s

( Since 127.9-127.4=0.5 s)

Also, the difference of Mean absolute deviation is: 0.03 s

( Since, 0.26-0.23=0.03 s)

Now let the difference of the mean times be equal t n times the difference of the mean absolute deviation of two teams

i.e.

0.5=n×0.03

⇒ n=0.5/0.03

⇒ n=16.6667

which is approximately equal to 16.

Hence, The difference between the mean times is about 16 times the mean absolute deviation of the data sets.

Flying to Kampala with a tailwind a plane averaged 158 km/h. On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. Also can I please get an explination/Walkthrough?

Answers

The answer will be 270 km/h hope this helps. :)

Which statement below best describes the transformation of parallelogram WXYZ to parallelogram W' X' Y' Z'?

A a reflection across the x axis
B a reflection across the y axis
C a reflection across the x axis and then across the y axis
D a translation of 8 units down

Answers

The answer is B. A reflection across the x-axis.

The answer is B <<<<=====
A reflection across the x axis means that any point is moved down the same number of squares as that point is above the x axis.

It is like seeing your reflection in a mirror. Your face looks like it is the same distance behind the glass as you measure your face in front of the glass.

what is the comon denominator of 3/8 and 1/3 as fractions for both

Answers

24 is a common denominator.

What is the comon denominator of 3/8 and 1/3 as fractions for both
its 12

unit price—- a long distance phone charge of $1.40 for 10 minutes, or $4.50 for 45 minutes —- show work^

Answers

Final answer:

To find the best long-distance phone charge, the unit prices for both options are calculated. The $1.40 charge for 10 minutes has a unit price of $0.14 per minute, while the $4.50 charge for 45 minutes is $0.10 per minute. Hence, the $4.50 option is better.

Explanation:

To determine the better long-distance phone charge, we can calculate the unit price for each option by dividing the total cost by the number of minutes.

For the $1.40 charge for 10 minutes:
Unit Price = Total Cost / Total Minutes
Unit Price = $1.40 / 10 minutes = $0.14 per minuteFor the $4.50 charge for 45 minutes:
Unit Price = Total Cost / Total Minutes
Unit Price = $4.50 / 45 minutes = $0.10 per minute

Comparing the two unit prices, the charge of $0.10 per minute for 45 minutes is the better option since the cost per minute is lower.

Plz help it’s the ANDREA ONe

Answers

Assuming that all measurements are the same, as the formula for volume of a box is simply

S^3. Make this expression equal 90 cubic inches. Then take the cube root to solve for s. The value of s are your measurements for the length width and height of the box.

S = 4.48 in.

A wave traveling at 230 m/sec has a wavelength of 2.1 meters. What is the frequency of the wave?

Answers

Velocity = wavelength * frequency,
frequency = velocity/wavelength,
230/2.1,
109.5 Hz

Hope this helps :)

Answer:

The frequency of the wave is 109.52 Hz.

Step-by-step explanation:

Given : A wave traveling at 230 m/sec has a wavelength of 2.1 meters.

To find : What is the frequency of the wave?

Solution :

We know that,

[tex]\text{Velocity} =\text{ Wavelength}\times \text{Frequency}[/tex]

We have given,

Velocity of wave is 230 m/s.

Wavelength is 2.1 meters.

Substitute in formula,

[tex]230 =2.1\times \text{Frequency}[/tex]

[tex]\text{Frequency}=\frac{230}{2.1}[/tex]

[tex]\text{Frequency}=109.52[/tex]

Therefore, The frequency of the wave is 109.52 Hz.

One hundred nautical miles equals about 185 kilometers. To the nearest kilometer, how far in kilometers is 195 nautical miles?

Answers

(d km)/(195 nm) = (185 km)/(100 nm)
d = 1.95*185 km ≈ 361 km

195 nautical miles is about 361 kilometers.

Answer:

361 is the correct answer

One jug of plant food treats an area of 135 square ft. How many jugs Would be needed to treat an area of 165 square yards (try writing equation)

Answers

Final answer:

To treat an area of 165 square yards, you would need 11 jugs of plant food.

Explanation:

To find out how many jugs would be needed to treat an area of 165 square yards, we need to convert the area to square feet and then divide by the area treated by one jug of plant food.

1 square yard = 9 square feet, so 165 square yards = 165 × 9 = 1485 square feet.

Now, since one jug treats an area of 135 square feet, the number of jugs needed is 1485 ÷ 135 = 11 jugs.

Learn more about Calculating Quantity here:

https://brainly.com/question/32195961

#SPJ12

Jay and Libby are working outside when they see a blimp flying over the neighborhood. Jay is mowing the lawn approximately 2,500 feet (ft) from Libby when they both see the blimp. Libby looks up from her gardening at an angle of while Jay looks up at an angle of 35°. Which distance, in feet, is closest to the vertical height of the blimp above the ground?

A. 1,751 feet
B. 2,331 feet
C. 7,027 feet
D. 7,535 feet

Answers

I found your complete question in other sources.
To solve this problem we make the following system of equations:
tan (35) = ((y) / (x + 2500))
tan (43) = ((y) / (x))
Resolving we have:
y = 7026.828214
See attached images for the solution of the system of equations.
Answer: The vertical height of the blimp above the ground is:
C. 7,027 feet

The vertical height of the blimp above the ground, which is seen by Jay and Libby while working outside, is 7027 ft.

What is right angle triangle property?

In a right angle triangle, the ratio of the opposite side to the adjacent side is equal the tangent angle between the adjacent side and hypotenuse side.

[tex]\tan\theta=\dfrac{b}{a}[/tex]

Here, (a) is the adjacent side, (b) is the opposite side.

Let the height of the blimp from the ground is y ft and the length of the ground to the Libby's position is x ft shown in the attached image below.

As the angle made by the Libby at the height of the blimp is 43 degrees. Therefore,

[tex]\tan43=\dfrac{y}{x}\\x=1.0724y[/tex] ......1

Here, Jay is mowing the lawn approximately 2,500 feet (ft) from Libby when they both see the blimp.

Libby looks up from her gardening at an angle of while Jay looks up at an angle of 35 degrees. Therefore,

[tex]\tan35=\dfrac{y}{x+2500}[/tex]

Put the value of x in the above equation from equation one as,

[tex]\tan35=\dfrac{y}{1.0724y+2500}\\0.7002(1.0724y+2500))=y \\y\approx7027 \rm \; ft[/tex]

Hence, the vertical height of the blimp above the ground, which is seen by Jay and Libby while working outside, is 7027 ft.

Learn more about the right angle triangle property here;

https://brainly.com/question/22790996

Please help and explain your answer.

Answers

Tan(theta) = o / a
Tan(theta) = x / 2.1
Tan(37) = x / 2.1
x = 2.1 * tan(37)
x = 2.1 * 0.7536
x = 1.58
x = 1.6 to the nearest tenth

How can you use the distributive property to find the product of 5 and 6723?

Answers

The DISTRIBUTIVE PROPERTY is when you distribute the number to certain numbers

The table shows the fraction of a book karen read saturday and sunday.What fraction of her book did she read on these two days?

Answers

ince we know the answer in terms of the amount of remaining pages, we first need to calculate how much of the book remains. This can be calculated as follows: 1/5 + 2/7 = (7+2*5)/35 = 17/35. Therefore, on Sunday the remaining pages is 1- 17/35 or 18/35. 5/9 of 18/35 is 2/7, so Lindaread 2/7 of the book ...

A class voted for either rollerblading, swimming, or biking as their favorite summer activity. If rollerblading got 4% percent of the vote and biking got 34% what percentage of the class voted for swimming?

Answers

62% would be your answer you would add the 4% then subtract it from 100

Determine the circumference of a circle with a radius of 4 units.
16 units
8 units
32 units
4 units

Answers

Final answer:

The circumference of a circle with a radius of 4 units can be calculated using the formula C = 2πr, which equals approximately 25.13 units when using π as 3.14159.

Explanation:

To determine the circumference of a circle with a radius of 4 units, we need to use the circumference formula:

C = 2πr

Where C represents the circumference, π (pi) is approximately 3.14159, and r is the radius of the circle.

π is a special mathematical constant that represents the ratio of the circumference of any circle to its diameter, and for most practical calculations, it can be approximated as 3.14. Therefore, plugging the radius (4 units) into the formula we get:

C = 2 x 3.14159 x 4

C = 8 x 3.14159

C = 25.13272 units

We can round this answer to two decimal places for simplicity, obtaining:

C = 25.13 units

So, the circumference of a circle with a radius of 4 units is approximately 25.13 units.

Final answer:

The circumference of a circle with a radius of 4 units is approximately 25.12 units.

Explanation:

The circumference of a circle can be found using the formula C = 2πr, where C is the circumference and r is the radius. In this case, the radius is given as 4 units, so we can substitute that value into the formula to find the circumference.

C = 2π(4) = 8π units.

Since π is approximately 3.14, we can approximate the circumference as:

C ≈ 8 x 3.14 = 25.12 units.

Therefore, the circumference of the circle with a radius of 4 units is approximately 25.12 units.

Saul flips a coin 20 Times and determines that the probability of flipping heads is 0.55. Is this experimental or theoretical probability

Answers

This is experimental. Saul has conducted an experiment to come to this solution

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