What is the sum of the first 51 consecutive odd positive integers?

Answers

Answer 1

We call:

[tex] a_{n} [/tex] as the set of the first 51 consecutive odd positive integers, so:

[tex]a_{n} = \{1, 3, 5, 7, 9...\} [/tex]

Where:
[tex]a_{1} = 1[/tex]
[tex]a_{2} = 3 [/tex]
[tex]a_{3} = 5 [/tex]
[tex]a_{4} = 7 [/tex]
[tex]a_{5} = 9 [/tex]
and so on.

In mathematics, a sequence of numbers, such that the difference between two consecutive terms is constant, is called Arithmetic Progression, so:

3-1 = 2
5-3 = 2
7-5 = 2
9-7 = 2 and so on.

Then, the common difference is 2, thus:

[tex]a_{n} = \{ a_{1} , a_{1} + d, a_{1} + d + d,..., a_{1} + (n-2)d+d\} [/tex]

Then:

[tex]a_{n} = a_{1} + (n-1)d[/tex]

So, we need to find the sum of the members of the finite series, which is called arithmetic series:

There is a formula for arithmetic series, namely:

[tex] S_{k} = ( \frac{a_{1} + a_{k}}{2} ).k[/tex]

Therefore, we need to find:
[tex] a_{k} = a_{51} [/tex]

Given that [tex]a_{1} = 1[/tex], then:

[tex]a_{n} = a_{1} + (n-1)d = 1 + (n-1)(2) = 2n-1 [/tex]

Thus:
[tex]a_{k} = a_{51} = 2(51)-1 = 101[/tex]

Lastly:

[tex] S_{51} = ( \frac{1 + 101}{2} ).51 = 2601[/tex]

Related Questions

hello can you please help me posted picture of question

Answers

Total number of people = 10

Number of women = 6

Probability of selecting 1st women = 6/10
Probability of selecting 2nd women = 5/9
Probability of selecting 3rd women = 4/8

Probability that are 3 selected candidates are women = 6/10 x 5/9 x 4/8 = 1/6

So, the correct answer is option A

I suck at math lol. Could someone give me the answers in detail? Picture attached! Thanks

Answers

11] Volume of cylinder is given by:
Volume=πr²h
r=1.7m, h=15 m
plugging in the values in our formula we get:
V=π(1.7)²×15
V=43.35π

Answer: 43.35π m³

12]
Volume of the oblique cylinder will be:
V=πr²h
r=4 in, h=8
thus
V=π×4²×8
V=128π in²

Answer: 128π in²

What is the slope of the line? A) -3 B) - 1/3 C) 1/3 D) 3

Answers

(1,1) and (2,4)

slope = (y2 - y1)/(x2 - x1)
slope = (4 - 1)/(2 - 1) = 3/1 = 3

answer
D) 3

C. 1/3 if I'm not mistaken should be the answer. You already could eliminate A. -3 and B. -1/3 since both are negatives. The line of this is pointing straight forward therefore D. 3 and C. 1/3 is left to choose from now it doesn't have a point on 3 that's how you know right away D can't be the solution. Hope I helped

Part A: Explain in words the mistake Juanita made. Part B: Solve the equation 3x + 6 = 24

Answers

she didnt divide the whole problem by 3 it would be x+3=8 then the rest of the work

If she was going to divide by 3 to eliminate the x term's coefficient, she needed to divide all other terms by 3 and she did not. She only divided two of the three terms by 3 and left out the 6.

3x + 6 = 24
divide all terms by 3

3x/3 + 6/3= 24/3

x + 2= 8
subtract both sides by 2

x= 6

CHECK:
3x + 6 = 24
3(6) + 6= 24
18 + 6= 24
24= 24

ANSWER: She did not divide all terms by 3 when eliminating the x term's coefficient. x= 6

Hope this helps! :)

Q9 Q3.) Solve the matrix equation 4X + 5A = B

Answers

Hi there!

See pictures below for the answer! My handwriting isn't that good. Let me know if you can't read it :)

Which values of x would make a polynomial equal to zero if the factors of the polynomial were (x-2) and (x-11)?

Answers

if the factors of the polynomial were (x-2) and (x-11), then if anyone of them is equal "0", a polynomial will be equal zero.
x-2=0
x=2
x-11=0
x=11

A polynomial will be equal 0 if x=2, or x=11.

Answer:

[tex]x=2\text{ (or) }x=11[/tex]

Step-by-step explanation:

We have been given factors of a polynomial as [tex](x-2)[/tex] and [tex](x-11)[/tex]. We are asked to find the values of x that would make the polynomial equal to zero.

We will use zero product property of polynomials to solve our given problem. The zero product property of polynomials states that if any of two factors of a polynomial is zero, then their product will be equal to zero.

Upon equating our given factors to zero, we will get:

[tex](x-2)(x-11)=0[/tex]

[tex](x-2)=0\text{ (or) }(x-11)=0[/tex]

[tex]x-2=0\text{ (or) }x-11=0[/tex]

[tex]x-2+2=0+2\text{ (or) }x-11+11=0+11[/tex]

[tex]x=2\text{ (or) }x=11[/tex]

Therefore, [tex]x=2\text{ (or) }x=11[/tex] will make the polynomial equal to zero.

What situation should be used to rewrite 16(x^3+1)^2 -22(x^3+1)-3=0 as a quadratic equation

Answers

[tex]16(x^3+1)^2-22(x^3+1)-3=0[/tex]

Use substitution: [tex]x^3+1=t[/tex]

[tex]16t^2-22t-3=0\\\\16t^2-24t+2t-3=0\\\\8t(2t-3)+1(2t-3)=0\\\\(2t-3)(8t+1)=0\iff2t-3=0\ \vee\ 8t+1=0[/tex]

[tex]2t-3=0\ \ \ |+3\\\\2t=3\ \ \ |:2\\\\t=\dfrac{3}{2}\\..............................\\8t+1=0\ \ \ \ |-1\\\\8t=-1\ \ \ \ |:8\\\\t=-\dfrac{1}{8}[/tex]

we're going back to substitution:

[tex]x^3+1=\dfrac{3}{2}\ \vee\ x^3+1=-\dfrac{1}{8}\ \ \ \ |subtract\ 1\ from\ both\ sides\ of\ the\ equations\\\\x^3=\dfrac{1}{2}\ \vee\ x^3=-\dfrac{9}{8}\\\\x=\sqrt[3]{\dfrac{1}{2}}\ \vee\ x=\sqrt[3]{-\dfrac{9}{8}}[/tex]

[tex]x=\dfrac{1}{\sqrt[3]2}\ \vee\ x=-\dfrac{\sqrt[3]9}{2}\\\\\boxed{x=\dfrac{\sqrt[3]4}{2}\ \vee\ x=-\dfrac{\sqrt[3]9}{2}}[/tex]

hello can you please help me posted picture of question

Answers

1/25, since there are 25 square in total and there is only one center square.

your answer would be 1/25 since there are 25 squares in all. The 1 as a denominator would come from the probability.

what is the work for 34,992 divided by 81

Answers

34992/81 or 81*x=34992 = 432

A square pyramid has a base with an area of 20 square meters, and its lateral faces have a slant height of x meters. Sydney is constructing a second square pyramid with the same size base, but the lateral faces of her pyramid have a slant height twice as long, 2x. Which statement best describes how the surface area of Sydney’s pyramid compares to the surface area of the original pyramid?

1. Sydney’s pyramid will have the same surface area because the .5 in the expression for the area of the triangular faces will make up for the slant height being doubled.
2. Sydney’s pyramid will have the same surface area because the slant height is not used when finding surface area.
3. Sydney’s pyramid will have a surface area that is exactly double the original pyramid’s because the slant height is used when finding the area of every lateral face.
4. Sydney’s pyramid will have a surface area that is greater than the original pyramid’s but not double the area because the slant height is not used when finding the area of the base.

30 POINTS

Answers

the surface area of a square pyramid with base length b and slant height x is
[tex]A=b^2+2bx[/tex]

note that b²=base area

so, if we say that her original pyramid's surface area is [tex]A_1=b^2+2bx[/tex], then the new one has a slant height of twice that, ie, we replace x with 2x and see what happens
[tex]A_2=b^2+2b(2x)[/tex]
[tex]A_2=b^2+4bx[/tex]
if we try to work [tex]A_1[/tex] back into there
[tex]A_2=b^2+2bx+2bx[/tex]
[tex]A_2=(b^2+2bx)+2bx[/tex]
[tex]A_2=A_1+2bx[/tex]

see our options
option 1 is wrong since the surface area increased by 2bx

option 2 is wrong since the surface area increased by 2bx, also we do use the slant height when finding surface area

option 3 is wrong because we got [tex]A_2=A_1+2bx[/tex] and not [tex]A_2=2(A_1)[/tex]

option 4 is correct since the new surface area is greater than the original by 2bx

answer is option 4

Answer:

Sydney’s pyramid will have a surface area that is greater than the original pyramid’s but not double the area because the slant height is not used when finding the area of the base.

Step-by-step explanation:

Hey can you please help me posted picture of question

Answers

Shifting left can be expressed as addition of a number to x.

So,

F(x) shifted to left will be F(x + c), where c is any constant

So,

G(x) will be of the form F(x+c) or (x + c)²

From the given options, only option B lists such an expression.

So the answer is option B

Can someone help with this math question

Answers

This is a problem of region. As shown in the figure, we have straight lines. We know that the equation for non-vertical lines is often given in the slope-intercept form by:
[tex]y=mx+b[/tex]

being m the slope of the line and b the y-intercept of it.

On the other hand, if x = 0 then y = b.

We will order the equations above without inequalities like this:

[tex]y=-x-6[/tex]

So, the G and J are straight lines with negative slope. We need to find the one that matches the inequality. Taking a point, for example (0, 0), we will analyze Figure J, so in the inequality:

[tex]0+0\ \textless \ -6[/tex]

This is false, so this statement doesn't match the Figure J. Therefore, the answer is G.

can someone plz help me!!!

Answers

6√25, ∛27, √64, ∛64

6*5, 3, 8, 4

30, 3, 8, 4

∛64 > ∛27> √ 64 > 6√25

What is the y-intercept of the quadratic y=-2x^2-4x-5 ?
(-5,0)
(0,-2)
(-1,3)
(0,-5)

Answers

y - intercept is f(0) then (0; m)

We have: [tex]y=-2x^2-4x-5\to f(0)=-2(0)^2-4(0)-5=-5[/tex]

Answer: (0; -5)

2.
Find the annual percentage rate, using the annual percentage rate table.
Amount Financed: $8,900

Finance Charge: $1,030.62

Number of Payments: 24

Answers

Amount Financed: $8,900
Finance Charge: $1,030.62
Number of Payments: 24

(Finance Charge)/(Amount Financed)*100$=($1,030.62)/($8,900)*100$
(Finance Charge)/(Amount Financed)*100$=(0.1158)*100$
(Finance Charge)/(Amount Financed)*100$=$11.58

In the row of number of Payments 24, we look for:
(Finance Charge)/(Amount Financed)*100$=$11.58, and we to which annual porcentage rate it corresponds in the first row

Answer: The annual percentage rate is 10.75%

Laura is completing a craft project that involves covering only the lateral surface of a cylindrical container with fabric. The cylinder has a height of 16.8 in. and a radius of 14.6 in. To the nearest square unit, how much fabric does she need for this project? Use a calculator.

Answers

The answer is 1561 in^2
The lateral surface area formula of a cylinder is LA = 2 pi r h

We are given
r=14.6 in
h=16.8 in
pi=3.14

LA = 2 * 3.14 * 14.8 * 16.8 about 1561 in^2

Answer:

1,541

Step-by-step explanation:

An apple pie with a circumference of 28.26 inches is cut into 6 equal-sized pieces. Two pieces of the pie are eaten. What is the area of the pie that is left? Select the correct answer. Use 3.14 for π.

Answers

C = πd
d = C / π
d = 28.26 / 3.14
d = 9 in

but r = d/2 = 9/2 = 4.5 in

A = π r^2
= 3.14 x (4.5^2)
= 3.14 x 20.25
= 63.585

divided into 6 equal
so 63.585 / 6 = 10.5975

but ate 2, so there are 4 pieces left
4 x 10.5975 = 42.39

answer
the area of the pie that is left is 42.36 in^2

hey user!!

your answer is here...

we know that circumference of circle = πd

given the circumference of the apple pie = 28.26 in.

therefore πd = 28.26 in.

==> 3.14 × d = 28.26 in.

==> d = 28.26/3.14

==> d = 9 in.

the diameter of the apple pie is 9 in.

therefore radius of the apple pie = 9/2 = 4.5 in.

area of the apple pie = πr²

== 3.14 × 4.5 × 4.5

= 63.585 in²

it's divided into 6 equal parts and two pieces are eaten. so the area of leftover pie = 63.585/6 × 4

= 63.585/3 × 2

= 21.195 × 2

= 42.39 in²

cheers!!

Solve equation -6/7m=-84

Answers

The first step for solving this equation is to determine the defined range.
[tex] \frac{-6}{7m} [/tex] = -84, m ≠ 0
Multiply both sides of the equation by 7m.
-6 = -588m
Now switch the sides of the equation.
-588m = -6
Divide bot sides of the equation by -588.
m = [tex] \frac{1}{98} [/tex], m ≠ 0
Lastly,, check if the solution is in the defined range (it is) to find your final answer.
[tex]m = \frac{1}{98} [/tex]
Let me know if you have any further questions.
:)

To solve the equation -6/7m = -84, multiply both sides by the reciprocal of -6/7, which is -7/6. The solution is m = 98.

To solve the equation -6/7m = -84, we need to isolate the variable m on one side of the equation. We can do this by performing the inverse operation of what is currently being done to m. Since m is being divided by -6/7, we can multiply both sides of the equation by the reciprocal of -6/7, which is -7/6, to cancel out the division on the left side.

The steps to solve the equation are:

Multiply both sides of the equation by -7/6:

-7/6 * (-6/7m) = -7/6 * -84

The left side simplifies to m, and the right side becomes m = 98.

So, the solution to the equation is m = 98.

Q is directly proportional to r. q is 76 when r is 20. work out q when r is 45.

Answers

q α r
q=kr
thus
k=q/r
when r=20, q=76
hence
k=76/20=
q=76/20r
when r=45 the value of q will be:
q=76/20×45
q=171

Answer:

q=171

Step-by-step explanation:

Hello there!

In this explanation k is constant and it means that q is some constant amount bigger than r.

q=kr

76=k×20

76÷20=k

k= 3.8

q=3.8r

q=3.8×45

q=171

Can anyone find the Surface Area of E?

Answers

The surface area is 26 units squared. I simply counted the number of sides!

what is a multiple of 6,9,18 between 61 and 107

Answers

There are two possible answers: 72 and 90.

We can find the answer by looking at the largest number among the three of them: 18. The only multiples of 18 between 61 and 107 are 72 and 90. We can check that both numbers are also multiples of 6 and 9:

90 : 6 = 15
90 : 9 = 10

72 : 6 = 12
72 : 9 = 8

Find an equation in standard form for the ellipse with the vertical major axis of length 16 and minor axis of length 4

Answers

Final answer:

Given the lengths of the major and minor axes of an ellipse, one can derive the standard form equation of the ellipse. In this example, the lengths were 16 and 4, yielding a standard form equation of x²/4 + y²/64 = 1.

Explanation:

The subject of this question is about finding the standard form equation for an ellipse given the lengths of the major and minor axes. To find the equation of the ellipse, we need to identify the semi-major axis (a) which is half the length of the major axis, and the semi-minor axis (b), which is half the length of the minor axis.

In this case, the lengths of the major and minor axes are 16 and 4 respectively, which gives us a semi-major axis of 8 and a semi-minor axis of 2. The standard form equation of an ellipse with a vertical major axis is given by:

(x-h)²/b² + (y-k)²/a² = 1

Where (h, k) is the center of the ellipse. In this case, since the center of the ellipse is at the origin (0,0), the equation becomes:

x²/4 + y²/64 = 1

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Olly drank 1/2 of a 16 fluid ounce bottle of juice how many cups of juice are left in the bottle explain

Answers

If Olly drank 1/2 of 16 fluid ounces, then what the question really means is that Olly drank 8 fluid ounces.

Remember the conversion: 8 fluid ounces = 1 cup. Since Olly drank 8 fluid ounces, then using the conversion, we can tell that Olly drank 1 cup of juice.

The total number of cups of juice that are left in the bottle are 8x.

What is division?

Division is a process of splitting a number into two or more equal parts.

Given is that Olly drank 1/2 of a 16 fluid ounce bottle of juice.

Assume that 1 fluid ounce of bottle can fill {x} cups. So, we can write -

16 fluid ounce bottle of juice can fill {16x} cups.

So, 1/2 x 16 fluid ounce of juice will fill {1/2 x 16x} cups.

{1/2 x 16x} cups

8x cups

Therefore, the total number of cups of juice that are left in the bottle are 8x.

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Which diagram shows the most useful positioning of a rectangle in the first quadrant of a coordinate plane? Answers are in the pictures.

Answers

use this one and that will be useful

me thinks you do not have to deal with all the other stuff. you jus tuse "a" for the height and "b" with the width

A diagram that shows the most useful positioning of a rectangle in the first quadrant of a coordinate plane is: D. diagram D.

What is a quadrant?

In Mathematics and Euclidean Geometry, a quadrant is the area that is occupied by the values on the x-coordinate (x-axis) and y-coordinate (y-axis) of a coordinate plane or cartesian coordinate.

This ultimately implies that, there are four (4) major quadrants in any graph and these include the following:

Quadrant 1: both x and y are positive.Quadrant 2: x is negative while y is positive.Quadrant 3: both x and y are negative.Quadrant 4: x is positive while y is negative.

In this scenario, we can logically deduce that the coordianetes of this rectangle can be easily calculated by using diagram D;

Length, b = |b - 0| = b

Width, a = |0 - a| = a

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determine the qotient of 7,684 and 78

Answers

Hello!

To determine the quotient of 7684 and 78, you'll need to divide 7684 by 78 or 7684/78.

You can do this by dividing 7684 by 78 in a calculator. Once you've done so, your quotient or answer, rounded to the nearest tenth, will be...

98.5

I hope this helps

Final answer:

The quotient of 7,684 and 78 is 98 with a remainder of 60, determined through long division.

Explanation:

To determine the quotient of 7,684 and 78, follow these steps:

Set up the division problem using long division with 7,684 as the dividend and 78 as the divisor.

Divide 78 into the first two digits of the dividend (76), which does not go evenly. Move to the first three digits (768).

Divide 78 into 768 to begin finding the quotient. 78 goes into 768 approximately 9 times because 78 x 9 = 702.

Subtract 702 from 768 to get a remainder of 66 and bring down the next digit from the dividend, which is 4, forming 684.

Divide 78 into 684. It goes 8 times because 78 x 8 = 624.

Write 8 next to the 9 in the quotient to update the quotient to 98 and subtract 624 from 684 to get a new remainder of 60. Since there are no more digits to bring down from the dividend, 60 is the final remainder.

The quotient is 98 with a remainder of 60.

HELP ASAP PLEASE
Given that ABC ~ DEC, find the value of x. If necessary, round your answer to two decimal places.

Answers

Diagram from: https://brainly.com/question/10173009
If these are similar triangles, this means that the ratio of corresponding sides is equivalent. In this case, AB corresponds to DE as CB corresponds to CE. In ratio and proportion form:AB / CB = DE / CE
x / 5 = (18 - x) / 25
5x = 18 - x
6x = 18
x = 3

the circumference of a circle is 37.68 what is the radius

Answers

the main formula for finding circumference is c=2piR
circumference//37.68=2(3.14)r =6.28 now divide 6.28 by both sides which will give you 6 therefore the radius is 6

What is is the equation of the line perpendicular to y = 3x-7 that contains the point (6,8)?

A. 1/3x - 7
B. 1/3x + 6
C. -1/3x - 7
D. -1/3x + 10

Answers

The slope of the perpendicular line will be the negative reciprocal of 3, the slope of the given line.
m = -1/(3) = -1/3

Only choices C and D have this slope. Only choice D goes through the given point. (-1/3)*6 +10 = -2 +10 = 8.

The appropriate selection is ...
D. (-1/3)x + 10

The equation of the line perpendicular to y = 3x -7 that passes through the point (6, 8) is Option D. y = -1/3x + 10. The slope of the perpendicular line is -1/3. Using point-slope form, we determine the final equation.

To find the equation of a line perpendicular to y = 3x - 7 that passes through the point (6, 8), we must determine the slope of the perpendicular line first.

The slope of the given line is 3. The slope of a line perpendicular to this one is the negative reciprocal of 3, which is -1/3.

We now use the point-slope form of the linear equation, which is:

y - y₁ = m(x - x₁)

Substitute m = -1/3 and the point (6, 8):

y - 8 = -1/3(x - 6)

Distribute -1/3:

y - 8 = -1/3x + 2

Add 8 to both sides:

y = -1/3x + 10

Thus, the equation is Option D. -1/3x + 10.

A triangle has side lengths of 10 centimeters, 2 centimeters, and c centimeters.

Enter a value to complete the inequality that describes the possible values for c, the length of the third side of the triangle.

HELP ASAP I COULD GET A PRESIDENTIAL AWARD FOR THIS!!!

Answers

Answer:

Step-by-step explanation:

The possible values of the third side of the triangle can be written as, 8 < c < 12.

What is Triangle?

A triangle is a two dimensional figure which consist of three vertices, three edges and three angles.

Sum of the interior angles of a triangle is 180 degrees.

Given a triangle.

Side lengths of the triangle are 10 centimeters, 2 centimeters, and c centimeters.

We have to find the possible values of c.

By the triangle Inequality Theorem,

10 - 2 < c < 10 + 2

8 < c < 12

Hence the possible values of c are 9, 10 and 11.

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james just bought a large container of ice cream the label says that there are 16 servings of ice cream write a formula to help james determine the number of grams on one serving of ice cream x=number of grams of ice cream in the container y=number of grams per serving

Answers

The correct formula that must be used is y=x/16. Variable 'x' represents the total mass of ice cream in the container. Mass of ice cream per serving is represented by the variable 'y'. The total mass of ice cream is divided by sixteen to determine the number of grams per serving.

Final answer:

To determine the number of grams per serving of ice cream, use the formula y = x / 16, where x is the total number of grams in the container and y is the number of grams per serving.

Explanation:

To determine the number of grams per serving of ice cream, we can set up a formula using the variables x and y. Since the container has 16 servings of ice cream, the total number of grams in the container can be represented as 16y (16 servings multiplied by the number of grams per serving).

So, the formula to determine the number of grams per serving (y) is: y = x / 16

For example, if the total number of grams in the container (x) is 800g, then the number of grams per serving (y) would be 800g / 16 = 50g.

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