Mathematics High School

PLEASE HELP I WILL GIVE BRAINLIEST FOR THE CORRECT ANSWER!!

Use mathematical induction to prove that 1 + 2 + 3 + … + n = (1/2) n (a1 + an).

Use the result to find the sum of 1 + 2 + 3 + ... + 500.

Answers

Answer 1

Answer:

See below for proof.

[tex]S_{500}=125250[/tex]

Step-by-step explanation:

Given arithmetic series:

1 + 2 + 3 + … + n

Therefore:

[tex]S_n=1+2+3+...+(n-2)+(n-1)+n[/tex]

[tex]S_n=1+(1+1)+(1+2)+...+(1+n-3)+(1+n-2)+(1+n-1)[/tex]

[tex]S_n=1+(1+1)+(1+2(1))+...+(1+(n-3)(1))+(1+(n-2)(1))+(1+(n-1)(1))[/tex]

Let:

a = first term = 1d = common difference = 1n = nth term

Therefore:

[tex]S_n=a+(a+d)+(a+2d)+...+(a+(n-3)d)+(a+(n-2)d)+(a+(n-1)d)[/tex]

Reverse the order:

[tex]S_n=(a+(n-1)d)+(a+(n-2)d)+(a+(n-3)d)+...+(a+2d)+(a+d)+a[/tex]

Add the two expressions for Sₙ:

[tex]2S_n=(2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)[/tex]

Therefore, the term (2a + (n – 1)d) has been repeated n times:

[tex]2S_n=n(2a+(n-1)d)[/tex]

Divide both sides by 2:

[tex]S_n=\dfrac{1}{2}n(2a+(n-1)d)[/tex]

[tex]S_n=\dfrac{1}{2}n(a+a+(n-1)d)[/tex]

Replace  a with  a₁  (first term) and a + (n – 1)d  with  aₙ (last term):

[tex]S_n=\dfrac{1}{2}n(a_1+a_n)[/tex]

To find the sum of the series 1 + 2 + 3 + ... + 500, substitute the following values into the formula:

a₁ = 1aₙ = 500n = 500

Therefore:

[tex]\implies S_{500}=\dfrac{1}{2}(500)(1+500)[/tex]

[tex]\implies S_{500}=250(501)[/tex]

[tex]\implies S_{500}=125250[/tex]

Related Questions

add 4x²y,-2xy²,-5xy²,3x²y plz explain​

Answers

4x^2 - 2xy^2
5xy^2 +
3x^2y
_____________
12x^5y^4-2xy^2

This is so because 4+5+3 is 12, then using laws of indices to add your x and y you get x^5 and y^4

To simplify your answer to the lowest you have it in the form of
3x^2y^2(4x^2y - 2xy^2)

If you multiply this as well you get the same answer I got with the addition

Determine the opposite of 3p^7-8p

Answers

Step-by-step explanation:

Maybe you should flip flop

6 grade math pls help

Answers

answer

measure of o is 75

and n is same as o as the sides are same

hope this helps you

360 degrees to the left

HELP MEEEEEEEEEEEE 3.0

Answers

Answer:

I think that J would be the answer , It is more reliable than the left ones .

Hope it was helpful , good luck ^^

J would be the correct answer

I NEED HELP!
The function f(x) is a quadratic function and the zeros of f(x) are - 2 and 5. The y-intercept of f(x) is 30. Write the equation of the quadratic polynomial in standard form.
f(x)=___________

Answers

Answer:

Step-by-step explanation:

First of all you have to put the zeros into factor form.

y1 = (x + 2)(x - 5)

Notice that the x changes sigh. You say that the zeros are -2 and  5. To get y1 to go to zero, you must make x the opposite sign of what you are given.

Now you have to expand the factored form to get the standard form

y = x^2 + 2x - 5x - 10

y = x^2 - 3x - 10

That's not the right answer.

f(0) = 30 but you have - 10 at the end so you have to multiply the factored form by - 3

-3(f(x)) = -3(x + 2)(x - 5)

-3(f(x)) = -3(x^2 - 3x - 10)

f(x) = -3x^2 + 9x + 30    

f(0) = -3(0)^2 + 9(0) + 30

f(0) = 30

The quadratic still has roots of (x +2)(x - 5) but that 3 makes the y intercept = 30.

See the graph below.

I need help fast please

Answers

Answer:

-2/3

Step-by-step explanation:

8^(x+2) =8^-2x

since the base are the same we deal with the exponent

x+2=-2x

x= -2/3

The low temperature at 6:45 A.M. was −22°F. At 3:15 P.M. the temperature was 5°F. What was the range of temperature for the day? A −27°F B −17°F C 17°F D 27°F

Answers

Answer:

D

Step-by-step explanation:

-22°F+27°F=5°F

have a nice day or evening

Answer:

D

Step-by-step explanation:

I took the test

Please need help you know this

Answers

Answer:

Rule: Output = 2 * input + 1

Right blank: Output  = - 1

Left blank: Input = 4

Step-by-step explanation:

Rule

Start with the rule

Output = 2 * input + 1

Let the output = y

Let the input = x

y = 2x + 1

Now

Right Blank

x = - 1

y = 2*-1 + 1

y = -2 + 1

y = - 1

That's the output for -2

Left Blank

The second blank asks you to solve for the input (x) The output (y) is 9

9 = 2x + 1                 Subtract 1 from both sides

9-1 = 2x                    Combine

8 = 2x                       Divide by 2

8/2 = 2x/2

x = 4

A playground is rectangular with a length of 12 miles. If the area of the playground is 58 square miles, what is its width? Input your answer as a fraction.

Answers

Answer:

4 5/6 miles

Step-by-step explanation:

Area of a rectangle = length x width

58 = 12 x width

to find width, divide both sides of the equation by 12

58/12 = 4 whole number and 10 would remain

4 10/12

to simply, divide numerator and denominator by 2

4 5/6 miles

PLEASE HELP MEEEEE PLEASE!

Answers

Answer: my best guess would be c

Step-by-step explanation: I hope you figure this out maybe use Socratic app that might help

Answer:

(-2,2)

Step-by-step explanation:

the red line is y = x + 4 ( we know this as x = -4 is the x intercept)

the green one is y = -x (it intercepts in origin)

so, we equal the two equations to each other to get the interception point:

x + 4 = - x

4 = -2x

x = -2

we substitute x = -2 in any of the equations we get

y = -2 + 4 = 2  or y = -(-2) = 2

so the point is (-2,2) which is option C.

A number "X" greater than 8

Answers

Answer:

X > 8

Step-by-step explanation:

Using the keywords greater than helps you.

John ran a race in 36 minutes. If each lap takes 9 minutes to run, write and solve an equation to find out how many laps John had to run.

SHOW YOUR WORK AND SOLVE PLEASE

Answers

Answer:

4 laps

Step-by-step explanation:

Solved using a proportion.

[tex]\frac{1}{9} = \frac{y}{36}[/tex]

9 × y = 1 × 36

9y = 36

9y ÷ 9 = 36 ÷ 9

y = 4

Margaret owns (5/6) of a section of land. She planted corn on (1/4) of her land. What fraction of the entire section is planted with Margaret’s corn?

Answers

Answer:

3/5

Step-by-step explanation:

subtract the fractions then turn the decimal into a fraction.

How do you work the problem 6y < 18 out

Answers

Answer:

y<3

Step-by-step explanation:

6y<18

y<18/6

y<3

I hope this helps

Answer: you can see what number times 6 gets you 18 so it would be y<3

Step-by-step explanation:

how do i solve this question

Answers

Answer:

18.18

Step-by-step explanation:

total sum of angle of quadrilateral is 360

70+60+ 7x-4+4x-3=360

160-7 +11x =360

153 +11x =360

11x=360-153

x=207/11

x=18.18

finding area of circle with radius , giving brainliest :)) pls help asap

Answers

Answer:

number 2 is the answer or 4*pi inches² or 12.6

Step-by-step explanation:

Can someone please help me

Answers

$9.50
You take each price of the games so, 1x2=2 2x2=4 2.25x2=4.5. Then add it up = 10.50. Since you already used ten you do 20-10.50 and get 9.50


If each square in the following floor plan represents 2.5 feet, what are the dimensions of the living room?

Answers

Twinkle twinkle little star how I wonder what you are

The median for the 5 numbers 2647, 1956, 2250, 2141, and x is 2141. What is the largest possible value for the number x?

Answers

Answer:

2140

Step-by-step explanation:

Median can be described as the number that lies in the middle of a set of arranged observations. the observations can be arranged in ascending or descending order

in a set of 5 numbers, the median would be the 3rd number

the median is 2141. there was to be 2 numbers before and after 2141

1956, x 2141, 2250, 2647

the largest possible value of x is 2140. this is the largest number that is less than 2140

On a number line, show all values of x for which |x|>3

Answers

It’s something that’s for sure good luck!

1. Mr. Jay has 24 students and 75% of them are boys. How many are boys?

Answers

Answer:

18

Heart if this helped

HELP ME IN NUMBER 1 AND 2

Answers

Answer:

Number is is A Number 2 is 77 trust ik math

1.7. In a Maths class, the girl to boy ratio is 7:5.
If there are 20 boys, how many girls are in the class?
​

Answers

Answer:

28 girls

Step-by-step explanation:

Let total students be ‘X’

No of girls = (X-20)

Girls/Boys=7/5=(X-20)/20

7(20) =5(X-20)

140=5X-100

5X=240

X = 240/5

X =48

So, Total Students are 48

No of girls = (X-20) = 48-20 = 28

PLEASE HELP I HAVE NO IDEA WHAT TO DO!!!

Answers

Answer: multiply and round I'm pretty sure tell me if u want me to multiply for u

Step-by-step explanation:

Freddy exercise 3.5 hours per day 4 days last week he burned 440 calories per hour while exercising how many calories did Freddie burn by exercising last week​

Answers

Answer:

6,160 calories

Step-by-step explanation:

total hours of exercising = 3.5 x 4 = 14

burns 440 calories per each hour

total calories burned is 440 x 14

Yo pls help me out, the question is on the picture.
ill mark brainliest if its correct. NO LINKS!

Answers

The correct answer is X = 13! I don’t know how to explain but I am 100% that is the correct answer!

HELP! I NEED THIS QUESTION ANSWERED ASAP!!!!!
HERE!

Angle X measures 53°
What type of angle is angle X?

Answers:

1 ) Obtuse

2) Right

3) Acute

Answers

Answer:

3 ) Acute

Step-by-step explanation:

Acute Angle - An angle less than 90 degrees.

Right Angle - An angle that is exactly 90 degrees.

Obtuse Angle - An angle more than 90 degrees and less than 180 degrees.

Straight Angle - An angle that is exactly 180 degrees.

Answer:

3)Acute

Step-by-step explanation:

any angle that is under a 90 degree angle or a right angle, is an acute angle. any angle that is above 90 degrees or a right angle is an obtuse andgle

how much markup is $75 to $180

Answers

Answer:

105

Step-by-step explanation:

75+105 =180

Bentley had $25 to spend on 5 paintbrushes for his art class. After buying them, Bentley had $5 left. How much did each of the paintbrushes cost?

Answers

Answer:

$4

Step-by-step explanation:

$25-$5=20

20 divided by 5 = 4

Plz help me with this one!

Answers

Step-by-step explanation:

[tex]y = x + 5 \: \: \: \: y = - 3x + 1 \\ y = x + 5 \: \: \: \: x + 5 = - 3x + 1 \\ y = x + 5 \: \: \: \: 4x = - 4 \\ y = x + 5 \: \: \: \: x = - 1 \\ y = 4 \: \: \: \: x = - 1[/tex]

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