prove that (n+5)^2-(n+3)^2 is a multiple of 4 for all positive integer values of n

Answers

Answer 1

Answer:

Proved

Step-by-step explanation:

Given

[tex](n+5)^2-(n+3)^2[/tex]

Required

Show that [tex](n+5)^2-(n+3)^2[/tex] is a multiple of 4

Expand each bracket

[tex](n+5)(n+5)-(n+3)(n+3)[/tex]

Open brackets

[tex]n^2 + 5x + 5x + 25 - (n^2 + 3x + 3x + 9)[/tex]

[tex]n^2 + 10x + 25 - (n^2 + 6x + 9)[/tex]

Open bracket

[tex]n^2 + 10x + 25 - n^2 - 6x - 9[/tex]

Collect Like Terms

[tex]- n^2 + n^2 - 6n + 10n + 25 - 9[/tex]

[tex]- 6n + 10n + 25 - 9[/tex]

[tex]4n + 25 - 9[/tex]

[tex]4n + 16[/tex]

Factorize

[tex]4(n + 4)[/tex]

Hence, the multiples of [tex](n+5)^2-(n+3)^2[/tex] are [tex]4[/tex] and [tex](n + 4)[/tex]


Related Questions

Solve the system of equations 4x – 9y = 5 and 8x - y = -7 by combining the
equations.

Answers

Answer:

x=2 and y=0.

Step-by-step explanation:

Answer:

[tex]x=-1,\\y=-1[/tex]

Step-by-step explanation:

We can write the following system of equations as given:

[tex]\begin{cases}4x-9y=5\\8x-y=-7\\\end{cases}[/tex].

Multiply the first equation by [tex]-2[/tex] then add both equations to solve for [tex]y[/tex]:

[tex]\begin{cases}-8x+18y=-10\\8x-y=-7\\\end{cases},\\17y=-17,\\y=\fbox{$-1$}[/tex].

Now plug in [tex]y=-1[/tex] into any equation to solve for [tex]x[/tex]:

[tex]8x-(-1)=-7,\\8x+1=-7,\\8x=-8,\\x=\fbox{$-1$}[/tex].

Verify that the solution pair [tex](-1,-1)[/tex] works [tex]\checkmark[/tex]

The athletic departments at 10 randomly selected U.S. universities were asked by the Equal Employment Opportunity Commission to state what percentage of their nursing scholarships were presently held by women. The responses were 5, 4, 2, 1, 1, 2, 10, 2, 3, 5.

Required:
Find the mean, median, mode, and geometric mean. Which is the most appropriate measure of central tendency? The least appropriate? Explain your answer. Is there an outlier?

Answers

Answer: Mean = 3.5 , median = 2.5, mode = 2, geometric mean = 2.74

Median is the most appropriate measure of central tendency.

The least appropriate = mean

Yes there is an outlier.

Step-by-step explanation:

Given responses : 5, 4, 2, 1, 1, 2, 10, 2, 3, 5.

First arrange them in increasing order, [tex]\sqrt[10]{1\times1\times2\times2\times2\times3\times4\times5\times5\times10 }\\\\=\sqrt[10]{24000} \approx2.74[/tex]

Its sum = 35

Mean of n observations = (Sum of observations) ÷ n

Mean = (35) ÷ 10

=3.5

Here n =10 (even)

Median = average of middle most numbers = [tex]\dfrac{2+3}{2}=\dfrac52=2.5[/tex]

Mode = most repeated number = 2    (thrice)

geometric mean = [tex]\sqrt[n]{x_1\times x_2\times.... x_n}[/tex]

10 is an outlier as it is very large as compare to other numbers.

When outlier is present in data , the median is the most appropriate measure of central tendency.

Mean affected badly by the outlier so it the least appropriate.

Enter a numerical expression for the word expression.
11 decreased by 8

Answers

A numerical expression is a mathematical statement involving only numbers and one or more arithmetic operation symbols+,−,×,÷. A numerical expression always represents a particular number. For example, the numerical expression 5 + 20 − 9 is equal to the number 16 when simplified.

find the range of the function

Answers

Answer:

a

Step-by-step explanation:

NEEED HELP QUICK PLS PLS pls pls pls .

Answers

Answer:

step 4 is correct because h-¹(x)=2x+5

aight mate can someone help me lol

Answers

Answer:-61/20

Step-by-step explanation:

Answer:

-3 1/20

Step-by-step explanation:

The denominators share a common multiple of 20.

-5/4 = -25/20

-9/5 = -36/20

-25/20 + -36/20 = -61/20 = -3 1/20

Which angle is congruent to p

Answers

Q is congruent to P because they’re both got the same x and y number the x is 5 and -5 and for y is 3 and -3.

Answer: angle q is congruent to angle p

Step-by-step explanation:

congruent means same size shape, ect....

What is the equation that represents this pattern

Thanks

Answers

t = 3f -1

hope this helps

Answer:

something like f(t)=t+3

Step-by-step explanation:

Which of the following is NOT equal to sin pi/3

A) sin 2pi/3
B) cos pi/6
C) cos 11pi/6
D) sin 4pi/3

Answers

Answer:

D)  [tex]sin(\frac{4\pi }{3} ) = sin(\pi +\frac{\pi }{3} ) = -sin\frac{\pi }{3}[/tex]

Step-by-step explanation:

A)

[tex]sin(\frac{2\pi }{3} ) = sin ( \pi - \frac{\pi }{3} ) = sin\frac{\pi }{3}[/tex]

B)

[tex]cos (\frac{\pi }{6} ) = cos ( \frac{\pi }{2} - \frac{\pi }{3} ) = sin(\frac{\pi }{3})[/tex]

C)

[tex]cos (\frac{11\pi }{6} ) = cos ( \frac{3\pi }{2} + \frac{\pi }{3} ) = sin(\frac{\pi }{3})[/tex]

D)

 [tex]sin(\frac{4\pi }{3} ) = sin(\pi +\frac{\pi }{3} ) = -sin\frac{\pi }{3}[/tex]

Please help me ASAP........

Answers

Answer:

More on supplies than she earned in sales.

Answer:

She spent 25 dollars from her pocket. No profit

Step-by-step explanation:

Determine whether the rational root theorem provides a complete list of all roots for the following
polynomial functions.
f(x) = 4x^2 - 25
g(x) = 4x^2 + 25
h(x) = 3x^2 - 25

Answers

Answer:

1) Yes

2) No, this polynomial has complex roots

3) No, this polynomial has irrational roots

Step-by-step explanation:

a. Rational roots theorem provide a complete list of all roots of

f (x) = 4x² - 25

b. Rational roots theorem does not provide a complete list of all roots of

g (x) = 4x² + 25

c. Rational roots theorem does not provide a complete list of all roots of

h (x) = 3x² - 25

What is Function?

A relation between a set of inputs having one output each is called a function.

Functions are given as;

f (x) = 4x² - 25

g (x) = 4x² + 25

h (x) = 3x² - 25

Now,

a. f (x) = 4x² - 25

To find the roots we can set up 0,

f (x) = 4x² - 25 = 0

4x² - 25 = 0

4x² = 25

Divide by 4,

x² = 25/4

Take square root both side,

x = √25/4

x = ±5/2

Thus, The function has rational roots.

So, Rational roots theorem provide a complete list of all roots.

b) g (x) = 4x² + 25

To find the roots we can set up 0,

4x² + 25 = 0

4x² = - 25

Divide by 4,

x² = - 25/4

Taking square root,

x = √-25/4

x = ±5i/2

Thus, The function has complex roots.

So, Rational roots theorem does not provide a complete list of all roots.

c) h (x) = 3x² - 25

To find the roots we can set up 0,

3x² - 25 = 0

3x² = 25

Divide by 3,

x² = 25/3

x² = 8.333

Take square root both sides,

x = √8.333

x = ±2.89

Thus, The function has irrational roots.

So, Rational roots theorem does not provide a complete list of all roots.

Therefore,

a. Rational roots theorem provide a complete list of all roots of

f (x) = 4x² - 25

b. Rational roots theorem does not provide a complete list of all roots of

g (x) = 4x² + 25

c. Rational roots theorem does not provide a complete list of all roots of

h (x) = 3x² - 25

Learn more about the rational root theorem visit:

https://brainly.com/question/25216212

#SPJ5

Add 3 5/12+2 5/9. Simplify the answer and write as a mixed number.

Answers

Answer:

5 35/36

Step-by-step explanation:

add the whole number: 3+2= 5

add the fractions: 5/12 = 5/9= 35/36

add the whole number and fraction

5 35/36 so x= 5 35/36

Camille buys $50 worth of books at Target. She has a coupon for 20% off her entire purchase. What is the total cost of the books after the coupon is applied?

Answers

Answer:

$40

Step-by-step explanation:

If something is 20% off divide the number by 5 then take that number and minus it from the original number

For example

50 ÷ 5 =10

50 - 10 = 40

I hope this helps!

A ball rolls 12 meters in 3 seconds. What is its speed?​

Answers

Step-by-step explanation:

speed = distance/time

= 12/3 m/s

= 4 m/s

4

V = s/t

= 12/3

=4

S = move

T = time

V = speed

10 pens for $8.99.
25 pens for $19.75

Answers

What is the question?

Step-by-step explanation:

Answer:

so you need to find the cost of one pen, which ever is cheaper pick that one

5) If f(x) = 3x + 4, for which value of x does f(x) = 22

Answers

Answer:

6

Step-by-step explanation:

22 = 3x+4

subtract 4 from both sides

18 = 3x

divide both sides by 3

6 = x

Choose the correct answer:

PLS HELP ASAP 30 points plsss

Answers

Answer:

[tex]\frac{x-4}{5x}[/tex]

Step-by-step explanation:

The equation;

[tex]\frac{x^{2} - 8x + 16}{5x}[/tex] ÷ x - 4

Factor;

[tex]\frac{(x-4)^{2}}{5x}[/tex] ÷ ( x - 4 )

Rewrite in fraction form, so that one can perform division

[tex]\frac{(x-4)^2}{5x} * \frac{1}{x-4}[/tex]

Simplify, cancel out like terms on opposite sides of the fraction bar

[tex]\frac{x-4}{5x}[/tex]

Arthur has a box which contains 6 black balls and n (n ≥ 2) white balls, then at random, he takes 3 balls from the box. If the probability of taking any ball is the same and Pₙ is the probability of taking 1 black ball and 2 white balls, what is the largest possible value for Pₙ?​

Answers

Answer:

11/68 or 0.1618

Step-by-step explanation:

Balls

Black - 6White - n

Options of getting 3 balls

1. White, black, black2. Black, white, black3. Black, black, white

Probability P(n) in each option

1. WBB

P(w) = 6/(n + 6)P(b1) = n/(n + 6 - 1) = n/(n + 5)P(b2) = (n - 1)/(n + 5 - 1) = (n - 1)/(n + 1)

P(n) =

P(w)P(b1)(P(b2) = 6/(n+6) × n/(n + 5) × (n - 1)/(n + 4) =6n(n - 1)/(n + 6)(n + 5)(n + 4)

2. BWB

P(b1) = n/(n + 6)P(w) = 6/(n + 6 - 1) = 6/(n + 5)P(b2) = (n - 1)/(n + 5 - 1) = (n - 1)/(n + 4)

P(n) =

P(b1)P(w)(P(b2) = n/(n+6) × 6/(n + 5) × (n - 1)/(n + 4) =6n(n - 1)/(n + 6)(n + 5)(n + 4)

3. BBW

P(b1) = n/(n + 6)P(b2) = (n - 1)/(n + 6 - 1) = (n - 1)/(n + 5)P(w) = 6/(n + 5 - 1) = 6/(n + 4)

P(n) =

P(b1)P(b2)(P(w) = n/(n+6) × (n - 1)/(n + 5) × 6/(n + 4) =6n(n - 1)/(n + 6)(n + 5)(n + 4)

Final equation is same for each case:

P(n) = 6n(n - 1) / (n + 6)(n + 5)(n + 4)

The easy way to find the maximum is to try the numbers or graph.

Both of the methods give the maximum integer n = 11 or n = 12

See attached graph

At both values n we get P(n):

P(11) = 6*11*10 / 15*16*17 = 11/68 = 0.1618 (rounded)P(12) = 6*12*11 / 16*17*18 = 11/68 = 0.1618 (rounded)

Rounded to three decimal places, what is the value of r for this data set?

Variable x 8 15 3 7 2 14
Variable y 15 21 6 12 3 20

Choices:
A. 0.950
B. 0.938
C. 0.880
D. 0.981

Answers

I think the answer is B not sure though

HHHHHHEEEEEELLLLLPPPPPP!!!!!!!!!

Answers

Step-by-step explanation:

Area of Glass surface

= Area of Parallelogram - Area of Rectangle

= (20in * 16in) - (4in * 16in)

= 256in².

Hence it is 256 square inches.

P.S. Question photo got changed, ignore this.

Answer: The area of the figure is 32 square

plz mark brainly

A production company spent 45
1
2 days filming its first movie. The​ company's second movie was filmed in 7
3
4 weeks. Find the ratio of the first​ movie's filming time to the second​ movie's filming time. Compare in weeks.

Answers

Answer:

26 /33

Step-by-step explanation:

Given that :

Number of days taken filming first movie = 45 1/2

Number of weeks spent filming second movie = 8 1/4

Concerning days to weeks ;

45 1/2 days = 91/2 days = 91/2 ÷ 7 ;

91 / 2* 1/7 = 91/14 = 6. 5 weeks

Hence,

Ratio of first week fuming to second week filming ;

6.5 weeks : 8 1/4 weeks

6 1/2 : 8 1/4

13 / 2 : 33 /4

13 / 2 * 4 /33

52 / 66

26 / 33

xavier made 22 more baskets than Jared this basketball season. Write an expression for the number of basket xavier made this season.​

Answers

Answer:

J+22=X

Step-by-step explanation:

J represents Jared and since Xavier made 22 more baskets than Jared you add 22 onto Jared's amount of baskets to figure out how many Xavier made.

Need answer ASAP. Will give brainliest

Answers

Answer:

b

Step-by-step explanation:

V 34

Answer:

B

Step-by-step explanation:

credit to the other person

HELP ME EASY THINKING PROBLEM 20 POINTS

Answers

do you move the 6 for the 2?

Step-by-step explanation:

I'm sorry if its wrong

have a good day!

What are the truth values of Q v ~R?

Answers

Answer:

ttft

Step-by-step explanation:

-3g = 9 what does g =

Answers

Answer:

g=-3

Step-by-step explanation:

-3g=9

divide through by-3

g=9/-3

g=-3

Answer:

Step-by-step explanation:

Here you go mate

Step 1

-3g=9  Equation/Question

Step 2

-3g=9  Simplify

-3g=9

Step 3

-3g =9   Divide by 3

answer

-3

Hope this helps

Find the 43rd term in the sequence: -6, -3, 0,...

Answers

Answer: 129

Step-by-step explanation: 43•3=129

Grinnell Glacier in Glacier National Park in the US covered about 142 acres in 2007 and was shrinking at a rate of about 4.4% per year.3 (a) Write a formula for the size, , of the Grinnell Glacier, in acres, as a function of years since 2007. (b) Use the model to predict the size of the glacier in the year 2020. (c) According to the model, how many acres of ice were lost from the glacier between 2007 and 2010

Answers

Answer:

Step-by-step explanation:

Glacier National park has the area in year 2007 = 142 acres

It's shrinking at a rate = 4.4% per year

a). Since, shrink in the size of the glacier is an exponential process,  

    Therefore, formula for the size of the glacier in 'n' years,

    A = 142(1 - 0.044)ⁿ

    A = 142(0.956)ⁿ

b). Difference between year 2007 and 2020 = 13 years

    By substituting n = 13 years in the formula,

    A = 142(0.956)¹³

        = 79.11 acres

    Therefore, size of the glacier in year 2020 = 79.11 acres

c). Difference between year 2007 and 2010 = 3 years

    By substituting n = 3 years in the formula,

    A = 142(0.956)³

    A = 124.07 acres

    Loss of the ice from the glacier = 142 - 124.07

                                                         = 17.93 acres

    Therefore, loss of the ice from the glacier in 3 years = 17.93 acres

Mrs. Carter puts a variety of wrapped chocolate candies into a bag. There are 5 silver-wrapped candies, 1 purple-wrapped candy, 2 striped candies, and 4 gold-wrapped candies. If 15 students select one candy at a time out of the bag without looking, and replace the candy after each draw, how many students would be expected to select a gold-wrapped candy from the bag?
A. 4 Students
B. 5 Students
C. 15 Students
D. 60 Students

Answers

Answer:

a

Step-by-step explanation:

it won't be 5

nor 60

and it won't be 15

What does 58/6 equal

Answers

Answer:

9.66666666667

Step-by-step explanation:

The fraction 58

6

is equivalent to 9 2

3

.

This fraction is a improper fraction once the absolute value of the top number or numerator (58) is greater than the absolute value of the bottom number or denomintor (6). So, the equivalent fraction is a mixed number which is made up of a whole number (9) and a proper fraction (2

3

).

The fraction 58

6

is equal to 58÷6 and can also be expressed as a decimal as 9.66667 and expressed as a percentage as 966.7%.

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