Solve the following by using mathematical Induction. For >/ 1
1^2+2^2+3^3+......+ n^2 = 1/6 n(n+1)(2n+[)

Answers

Answer 1

Answer:

See explanation

Step-by-step explanation:

Prove that

[tex]1^2+2^2+3^3+...+n^2=\dfrac{1}{6}n(n+1)(2n+1)[/tex]

1. When [tex]n=1,[/tex] we have

in left part [tex]1^2=1;[/tex] in right part [tex]\dfrac{1}{6}\cdot 1\cdot (1+1)\cdot (2\cdot 1+1)=\dfrac{1}{6}\cdot 1\cdot 2\cdot 3=1.[/tex]

2. Assume that for all [tex]k[/tex] following equality is true

[tex]1^2+2^2+3^3+...+k^2=\dfrac{1}{6}k(k+1)(2k+1)[/tex]

3. Prove that for [tex]k+1[/tex] the following equality is true too.

[tex]1^2+2^2+3^3+...+(k+1)^2=\dfrac{1}{6}(k+1)((k+1)+1)(2(k+1)+1)[/tex]

Consider left part:

[tex]1^2+2^2+3^2+...+(k+1)^2=\\ \\=(1^2+2^2+3^3+...+k^2)+(k+1)^2=\\ \\=\dfrac{1}{6}k(k+1)(2k+1)+(k+1)^2=\\ \\=(k+1)\left(\dfrac{1}{6}k(2k+1)+k+1\right)=\\ \\=(k+1)\dfrac{2k^2+k+6k+6}{6}=\\ \\=(k+1)\dfrac{2k^2+7k+6}{6}=\\ \\=(k+1)\dfrac{2k^2+4k+3k+6}{6}=\\ \\=(k+1)\dfrac{2k(k+2)+3(k+2)}{6}=\\ \\=(k+1)\dfrac{(k+2)(2k+3)}{6}[/tex]

Consider right part:

[tex]\dfrac{1}{6}(k+1)((k+1)+1)(2(k+1)+1)=\\ \\\dfrac{1}{6}(k+1)(k+2)(2k+3)[/tex]

We get the same left and right parts, so the equality is true for [tex]k+1.[/tex]

By mathematical induction, this equality is true for all n.


Related Questions

Write a ratio as a ratio of whole numbers in lowest terms
$1.00 to $1.50

Answers

Answer:

Step-by-step explanation:

50

The ratio of the whole numbers in its lowest term is ratio $2 to $3 i.e.  [tex]\mathbf{\dfrac{2}{3}}[/tex]

A ratio of two integers a and b can be represented into a fraction as numerators and denominators.

[tex]\mathbf{\dfrac{a \to numerator}{b \to denominator}}[/tex]

To recognize and determine the ratio of the whole numbers in their lowest terms, we will multiply both the numerator and the denominator by 10.

i.e.

[tex]\mathbf{=\dfrac{1.00}{1.50}}[/tex]

[tex]\mathbf{=\dfrac{1.00\times 10}{1.50 \times 10}}[/tex]

[tex]\mathbf{=\dfrac{10}{15}}[/tex]

DIvide both the numerator and denominator by 5;

[tex]\mathbf{=\dfrac{2}{3}}[/tex]

Therefore, we can conclude that the ratio of the whole numbers in its lowest term is ratio $2 to $3 i.e.  [tex]\mathbf{\dfrac{2}{3}}[/tex]

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Graph f(x)=1/2x+2. Use the line tool and select two points to graph the line.

Answers

Graph f(x)=1/2x+2. Use the line tool and select two points to graph the line.

Step-by-step explanation:

Since f is a linear function then its graph must be a line

to draw a line ,all you need is plotting two points.

f(0)=1/2(0)+2=0+2=2 then the line passes through the point (0,2)

f(2)=1/2(2)+2=1+2=3 then the line passes through the point (2,3)

look ok at the photo below for the graph.

Which is a common denominator of 1/4 and 3/5 ?

A. 9

B. 15

C. 16

D. 20
I WILL MARK BRAINLIST IF CORRECT !! :)

Answers

Answer:

D

Step-by-step explanation:

5*4=20

4*5=20

Skyler is 104 miles away from Marcus. They are traveling towards each other. If Marcus travels 8 mph faster than skyler and they meet after 4 hours, how fast was each traveling?

Answers

Speed of skyler is 204 mph and speed of marcus is 212 mph

Solution:

Skyler is 104 miles away from Marcus

Therefore,

Distance = 104 miles

They meet after 4 hours

Time taken = 4 hours

Let "x" be the speed of sklyer

Marcus travels 8 mph faster than skyler

Therefore,

Speed of marcus = 8 + x

They are traveling towards each other

Since they are traveling towards each other, we can add up their speeds

Speed = x + 8 + x = 2x + 8

The distance is given by formula:

[tex]distance = \frac{speed}{time\ taken}\\\\104 = \frac{8+2x}{4}\\\\104 \times 4 = 8 + 2x\\\\8 + 2x = 416\\\\2x = 416-8\\\\2x = 408\\\\x = 204[/tex]

Thus speed of skyler = x = 204 mph

Speed of marcus = 8 + x = 8 + 204 = 212 mph

Thus speed of skyler is 204 mph and speed of marcus is 212 mph

Typically, what percentage of the home cost should you have available for a down payment? 5%, 50%, 25%, 30%

Answers

Answer:

25%

Step-by-step explanation:

Answer:

25%

Step-by-step explanation:

I had this question on a test and i made an 100%

Mrs. Rivera ate 3/8 of a
pizza and Ms. Carter ate 4/8
of a pizza. How much pizza
did they eat altogether?

Answers

Answer:

They ate 7/8 of a pizza altogether.

Step-by-step explanation:

So, you know Mrs. Rivera ate 3/8 of a pizza and Ms. Carter ate 4/8 of a pizza.

You add 3/8 + 4/8 and you get 7/8.

The ate 7/8 of a pizza altogether.

The quantity of pizza  that Mrs. Rivera and Ms. Carter ate altogether is equal to [tex]\frac{7}{8}[/tex]

Given the following data:

Fraction Rivera ate = [tex]\frac{3}{8}[/tex]Fraction Carter ate = [tex]\frac{4}{8}[/tex]

To calculate how much (quantity) pizza  they ate altogether:

A fraction can be defined as a numerical quantity that isn't a whole number.

Hence, a fraction is simply a part of a whole number.

In this exercise, we would add up the two fractions, so as to calculate how much (quantity) pizza  they ate altogether.

[tex]Total \;quantity = \frac{3}{8} + \frac{4}{8}[/tex]

Lowest common multiple (LCM) = 8

[tex]Total\;quantity = \frac{4\;+\;3}{8} \\\\Total\;quantity = \frac{7}{8}[/tex]

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which point is an x intercept of the quadratic function f(x) =(x-8)(x+9)

Answers

Answer:

(- 9, 0) and (8, 0)

Step-by-step explanation:

To find the x- intercepts let f(x) = 0, that is

(x - 8)(x + 9) = 0

Equate each factor to zero and solve for x

x + 9 = 0 ⇒ x = - 9

x - 8 = 0 ⇒ x = 8

The x- intercepts are x = - 9 and x = 8

There are two functions that can be used to describe the percent of people 25 years and older who have completed at least high school. For both functions f(x)=0.44x + 87.5 and g(x)=0.008x^2 + 0.42x + 87.5 is the number of years since 2008, and any y (or f(x) or g(x)) is the percent of people 25 and older completing at least high school. Assume that the trend of g(x) continues. Find g(15) and describe in words what this means.

g(15)=

Answers

Answer:

[tex]g(15)=93.8\%[/tex]

see the explanation

Step-by-step explanation:

Let

x ----> the number of years since 2008,

g(x) ----> the percent of people 25 years and older who have completed at least high school

we know that

[tex]g(x)=0.008x^2+0.42x+87.5[/tex]

so

For x=15 years since 2008

substitute

[tex]g(15)=0.008(15)^2+0.42(15)+87.5=93.8\%[/tex]

That means

In the year 2023 (2008+15) the percent of people 25 and older completing at least high school will be 93.8%

Final answer:

To find g(15), we substitute x with 15 in the function g(x) = [tex]0.008x^2 + 0.42x + 87.5[/tex], yielding g(15) = 95.6%. This means that in 2023, it's projected that 95.6% of people 25 and older will have completed high school according to the given quadratic trend.

Explanation:

The student is asking for the evaluation of the function g(x) = [tex]0.008x^2[/tex] + 0.42x + 87.5 at x = 15, which represents the number of years since 2008. To find g(15), we need to substitute x with 15 in the given function:

g(15) = 0.008(15)2 + 0.42(15) + 87.5

Calculating the value step by step:

152 = 225

0.008 × 225 = 1.8

0.42 × 15 = 6.3

1.8 + 6.3 + 87.5 = 95.6

Therefore, g(15) = 95.6.

In words, this means that if the trend described by function g(x) continues, then in the year 2023 (15 years after 2008), 95.6% of people aged 25 and older will have completed at least a high school education. This is a projection of graduation rates over time.

Jake paid $13.50 for admission to the country fair and bought 9 tickets to play games. If he spent a total of $36, what is the cost, c, of one ticket? Write and solve an equation.

Answers

Answer:

  $2.50

Step-by-step explanation:

Jake's total expenditure is ...

  total = admission + c × (number of tickets)

  36 = 13.50 + 9c . . . an equation to solve

  4 = 1.50 +c . . . . . . . divide by 9

  2.50 = c . . . . the cost of one ticket

Each ticket cost Jake $2.50.

Daniel is constructing a deck like the one shown. What is the area of the deck? Explain how you found the answer.

Answers

Answer:

Well if the deck was sown i would know butt... ill just give an example so if the height is 5 and the length 10 then you do 5 times 10 and get 50 A=50

Step-by-step explanation:

Answer:

119.5

Step-by-step explanation:

um....well i looked it up so yeah

Round to 2 significant figures - 0.002353

Answers

Answer:

Step-by-step explanation:

Dave works for 5 hours double time and he earns $98 find his normal hourly year rate

Answers

Answer: $9.8 per hour

Step-by-step explanation:

If Dave works for 5 hours at double time, then he gets paid for 10 hours of work. If he earned $98 for 10 hours of work, all you need to do is divide the amount of money earned by the hours of work. [tex]\frac{98}{10}[/tex]=9.8 dollars per hour.

given S (-1,8), and T (7,-9) what is the length of ST?

Answers

Answer:

ST = √353 or 18.79

I can add a step-by-step explanation if you want me to.

The length of ST is approximately 18.79 units.

What is the distance between two points ( p,q) and (x,y)?

The shortest distance (length of the straight line segment's length connecting both given points) between points ( p,q) and (x,y) is:

[tex]D = √[(x-p)² + (y-q)²] D = \sqrt{(x-p)^2 + (y-q)^2} \: \rm units.[/tex]

We are given that;

Points= S (-1,8), and T (7,-9)

Now,

x1​=−1

y1​=8

x2​=7

y2​=−9

Substituting these values into the formula, we get:

d=(7−(−1))2+(−9−8)2​

d=(8)2+(−17)2​

d=64+289​

d=353​

d= 18.79

Therefore, by the distance the answer will be 18.79 units.

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F is directly proportional to a.
If F = 24 when a = 8 find,
a when F = 18​

Answers

Answer:

K = 3

a = 6

Step-by-step explanation:

F is directly proportional to a

That’s F = Ka

K is the constant

Divide both sides by a to get the constant k

K = F/a

When F = 24 and a =8

K = 24/8

= 3

When F = 18

a = ?

F = Ka

18 = 3 x a

18 = 3 x a

Divide both sides by 3

18/3 = 3 x a /3

6 = a

a = 6

How can this 12×2+2³ make 120

Answers

Step-by-step explanation:

[tex]12\times(2+2^3)\qquad\text{use PEMDAS}\\\\\text{first power}\\\\=12\times(2+8)\\\\\text{next addition in parentheses}\\\\=12\times10=120[/tex]

In 2000 the median home price in a certain city was about $290,000, and from 2000 to 2006, home prices in the city rose in an average of about 11.3% per year. If prices continue to rise at that rate what would be the median home price would have been in 2018? Compare to the actual median price of about $400,000 in 2018. (median home price would be approximately ____ minion in 2018.

Answers

Answer:

5000

Step-by-step explanation:

How to solve systems using elimination

Answers

See explanations below on solving simultaneous equations by elimination.

Step-by-step explanation:

In solving simultaneous equations by elimination, you make one side of the variables equal and eliminate that side by either addition or subtraction, then solve the remaining variable whose value is then used in one of the equations to find the other variable. For example;

(i)   4x - 2y= 10

(ii)  x+ y =4

Making the x variable part equal to eliminate x

1(4x-2y=10)--------multiply by 1

4(x+y=4) ---------multiply by 4

4x -2y =10

4x+4y=16

Apply subtraction to eliminate the x variables

0 +6y =6

6y=6

y=6/6 = 1

Use the value of y =1 in the equation x+y= 4

x+y=4

x+1=4

x=4-1 = 3

x= 3,  and y=1

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Two people working together can complete a job in 5 hours. If one of them works
twice as fast as the other, how long would it take the faster one working alone?​

Answers

Answer:

5 hours

Step-by-step explanation:

because he is working twice as fast which means he will get the job done faster

5 hours times 2=10- 5 =5
So your awnser would therefore be 5

Find the measure of angle EPH Enter your answer in the box.

Answers

Answer:

50

Step-by-step explanation:

list the sample space for odd numbers between 20 and 30

Answers

The odd number between 20 and 30 are 21,23,25,27,30
Final answer:

The sample space for odd numbers between 20 and 30 is {21, 23, 25, 27, 29}. These are integers that are not wholly divisible by 2.

Explanation:

The sample space is a term used in mathematics, specifically in statistics and probability, to describe all possible outcomes of an experiment. In this case, the experiment is to find all odd numbers between 20 and 30. An odd number is an integer that is not exactly divisible by 2. The odd numbers between 20 and 30 are: 21, 23, 25, 27, and 29. Therefore, the sample space you are looking for is {21, 23, 25, 27, 29}.

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The Outstanding O's cereal box has been redesigned with a new size and shape.
20 cm
30 cm
8 cm
How much material will it take to make the new box with no overlaps?

Answers

Answer:

Step-by-step explanation:      

2(20x30)+ 2(8x30) + 2(20x8)                                                                                                      -                                                                                                                                                                                                                                               I                                                                                                                                                                                                                >          1200 + 480 + 320=                                                                                                       2000                                                                                                                                  hope this helps :D

The material required for making a new cereal box with new dimensions is 2000 sq. cm. This is obtained by calculating the total surface area of the box with new dimensions.

The surface area of the cuboid:Cuboid has 6 faces (each face looks like a rectangle)To get the surface area of the cuboid, add the areas of all the 6 faces of the cuboid.For the length (L), width (W), and height (H), the surface area is given as S.A=2(LW + WH + LH). Since the area of the rectangle is L×W, adding areas of each face.

Calculating the required material:

It is given that,

The Outstanding O's cereal box has been redesigned with the new size and shape

The new dimensions are,

L=20 cm

W=8 cm

H=30 cm

To get the required material, the surface area of the box with new dimensions is to be calculated.

So,

S.A=2(LW + WH + LH)

     =2(20×8+8×30+30×20)

     =2(160+240+600)

     =2(1000)

     =2000 sq. cm

Therefore, the material required for the new box is 2000 sq. cm

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What is slope = 3/4 goes through the point (0,6)

Answers

Answer:

y=3/4x+6

Step-by-step explanation:

y-y1=m(x-x1)

y-6=3/4(x-0)

y-6=3/4(x)

y-6=3/4x

y=3/4x+6

The distance a car can travel in miles is modeled by the equation y=23x-6, where x represents the number of gallons of gas the car uses. If the car traveled 86 miles, how many gallons of gas did it use

Answers

Answer:

4 gallons

Step-by-step explanation:

86 = 23x -6

92 = 23x

x = 4

Answer:

4

Step-by-step explanation:

y=23x-6

86=23x-6

92=23x

x=92/23

x=4

Combine like terms to create an equivalent expression 2/5m - 4/5 - 3/5m

Answers

Answer:

-1/5m-4/5

Step-by-step explanation:

2/5m-4/5-3/5m

2/5m-3/5m-4/5

-1/5m-4/5

what is the measurement of

Answers

Answer: Angle A = 120° and angle C = 45°

Step-by-step explanation: The first thing to note at the back of our minds is that the sum of the interior angles of a triangle equals 180°. Given that the three angle are available, we can start by adding them all together. Hence,

{12x + 12} + 15 + {3x + 18} = 180

12x + 12 + 15 + 3x + 18 = 180

By collecting like terms, we arrive at,

12x + 3x + 12 + 15 + 18 = 180

15x + 45 = 180

Subtract 45 from both sides of the equation

15x = 135

Divide both sides of the equation by 15

x = 9

At this point, we can now substitute for the value of x = 9 in the expressions representing angles A and C.

Angle A = 12x + 12

= 12(9) + 12

= 108 + 12

= 120

Angle A = 120°

Angle C = 3x + 18

= 3(9) + 18

= 27 + 18

= 45

Angle C = 45°

A 500-kilogram horse runs at 5 meters/second. The horse’s kinetic energy is (blank)joulese g = 9.8 N/kg.)

Answers

Answer:

KE = 6250 Joule.

Step-by-step explanation:

Every moving body has kinetic energy and it is given by the expression

[tex]KE = \frac{1}{2}mv^{2}[/tex]

where the moving body has m as the mass and v as the velocity.

If m is in Kg and v is in meter/sec, then KE will be expressed in Joule.

Now, in this case, m = 500 and v = 5 meter/sec.

So, [tex]KE = \frac{1}{2} \times 500 \times (5)^{2} = 6250[/tex] Joule. (Answer)

Rita has n dollars yo being with. She just spent 23 dollars. Using n , write an expression for the number of dollars she has left.

Answers

answer:

n - 23 = x



I’m not entirely sure but this is my understanding.

Answer:

i think n-23=x

the question is worded weirdly

I need to know the answers to these please

Answers

Find least common multiple of 8 and 10.
The answer is 40.

7/8=x/40. We had to multiply 8 by 5 to get 40.

Whatever you do to denominator you must do to numerator. Multiply 7 and 5 to get 35.

Now we have 7/8 is equal to 35/40.

Next, for the 9/10 part. We multiplied 10 and 4 to get 40 so multiply the numerator by 4 as well and you get 36.

Now we have 9/10 is equal to 36/40

The problem is now 35/40 and 36/40

35/40 < 36/40

Which is the same as

7/8 < 9/10

Write the equation.

During the year,

Myong travels 5,678

miles by airplane.

He travels twice as

many miles by train.

How many miles

does Myong travel

by train?

Answers

Answer:

11,356

Step-by-step explanation:

We multiply 5,678 by 2 because the word 'twice' is the same as multiplying by 2

Myong travels 11356 miles by train because 5678 x 2 is 11356

Rewrite the following in the form log(c).
log(6) – log(2)

Answers

Answer:

Step-by-step explanation:

㏒(6)-㏒(2)=㏒(6/2)=㏒ 3

The expression log(6) – log(2) in form of log(c) is log(12).

What is Logarithm?

Logarithms are another way of writing exponents in mathematics. A number's logarithm with a base equals another number. Exponentiation is the inverse function of logarithm.

A logarithm is defined as the number of powers to which a number must be increased in order to obtain some other numbers. It is the simplest way to express enormous numbers.

We have to write the expression log(6) – log(2) in form of log(c).

Using product property for logarithms

log a + log b= log (ab)

Here, a= 6 and b= 2 then

ab = 6 . 2= 12

Thus, the required form of log(6) – log(2) is log(12).

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