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

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

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]

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

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.

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

Answers

Answer:

18

Heart if this helped

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.


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

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

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!

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

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

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

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:

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

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

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

A number "X" greater than 8

Answers

Answer:

X > 8

Step-by-step explanation:

Using the keywords greater than helps you.

HELP ME IN NUMBER 1 AND 2

Answers

Answer:

Number is is A Number 2 is 77 trust ik math

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:

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

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

Answers

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

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

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.

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

how much markup is $75 to $180

Answers

Answer:

105

Step-by-step explanation:

75+105 =180

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

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

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