Mathematics High School

Write the sum using summation notation, assuming the suggested pattern continues.

2 - 12 + 72 - 432 + ...

Answers

Answer 1
Answer: [tex]\displaystyle \sum_{k=1}^{n}2(-6)^{k-1}[/tex]

The first term is a = 2. The common ratio is r = -6. You start off with 2 and multiply each term by -6 to get the next term. 

eg: 2*(-6) = -12 ---> -12*(-6) = 72 etc etc

If you want the series summation to go on forever, then replace the 'n' up top with the infinity symbol [tex]\infty [/tex]

Note: Because |r| < 1 is not true (r < 1 in this case), this means that the series diverges.
Answer 2

Answer:

[tex]\displaystyle \sum_{k=1}^{n}2(-6)^{k-1}[/tex]

Step-by-step explanation:

we have to write the sum using summation notation of the following series:

2-12+72-432+...

which could also be written as:

2+(-12)+72+(-432)+...

We observe that first term=2

second term=2×-6

Third term=2×-6×-6

Fourth term=2×-6×-6×-6

nth term=[tex]2\times (-6)^{n-1}[/tex]

Hence, the series sum=   [tex]\displaystyle \sum_{k=1}^{n}2(-6)^{k-1}[/tex]

Related Questions

Kenya plans to make a down payment plus monthly payments in order to buy a motorcycle. At one dealer she would pay $2,500 down and $150 each month. At another dealer, she would pay $3,000 down and $125 each month. After how many months would the total amount paid be the same for both dealers? What would that amount be?

Answers

Let us say that the total amount paid in the first dealer is P1 and the total amount paid to the second dealer is P2. So that:

P1 = 2500 + 150 t

P2 = 3000 + 125 t

Where t is the total number of months

Now we are asked when the total amount paid would be equal for the two dealers, this means P1 = P2, therefore equating the two:

2500 + 150 t = 3000 + 125 t

25 t = 500

t = 20 months

 

Therefore the total amount paid for both dealers would be equal after 20 months.

A lawn mower manufacturer incurs a total of $34,816 in overhead costs and $388 per lawn mower in production costs. How many lawn mowers were manufactured if the average cost of production is $660?

Answers

There are 128 law mowers manufactured. 

I hoped I helped and if you need more you could always ask me :)

-Dawn

Answer:

128 lawn mowers

Step-by-step explanation:

Given,

The overhead cost = $ 34,816,

The cost for one lawn mower = $ 388,

Let x be the number of lawn mowers manufactured,

So, the total cost of x lawn mowers = 34816 + 388x,

Now, if the average cost of x mowers = $ 660,

So, the total cost = 660x

[tex]\implies 660x = 34816 + 388x[/tex]

[tex]660x - 388x = 34816[/tex]

[tex]272x= 34816[/tex]

[tex]\implies x = \frac{34816}{272}=128[/tex]

Hence, 128 lawn mowers were manufactured.

HELP!!!!!!! I GIVE BRAINLEST AND THANKS!!!!!! + 5 POINTS!!!!

Answers

In order for a relation to be a function, each x value must have no more than 1 corresponding y value.

Remember that the x values are the ones on the left in an ordered pair. Because each one is unique, the relation is a function in this case.

Final answer: B

Four times the sum of a number and 15 is at least 120. Find all possible solutions for x

Answers

First translate the English phrase "Four times the sum of a number and 15 is at least 120" into a mathematical inequality.

"Four times..." means we're multiplying something by 4.

"... the sum of a number and 15..." means we're adding an unknown and 15 and then multiplying the result by 4.

"... is at least 120" means when we substitute the unknown for a value, in order for that value to be in the solution set, it can only be less than or equal to 120.

So, the resulting inequality is 4(x + 15) ≤ 120.

Simplify the inequality.

4(x + 15) ≤ 120
4x + 60 ≤ 120 <-- Using the distributive property
4x ≤ 60 <-- Subtract both sides by 60
x ≤ 15 <-- Divide both sides by 4

Now that we have the inequality in a simplified form, we can easily see that in order to be in the solution set, the variable x can be no bigger than 15.

In interval notation it would look something like this:

[15, ∞)

In set builder notation it would look something like this:

{x | x ∈ R, x ≤ 15}

It is read as "the set of all x, such that x is a member of the real numbers and x is less than or equal to 15".

EF=5x+15, FG=53 And EG=143 find the value of x
And can you please tell me how to do it?

Answers

I'm assuming EFG is a triangle. In order to solve this problem, you do not have all the information to solve for x in a numerical form. However, if you let variable P equal the perimeter. You can solve for x in terms of P. Perimeter is the sum of all the sides of a shape. First, let's add together all the sides of the triangle. (P = 5x+15+53+143). Add together the common variables so that (5x+211=P). Our next job is to isolate the x. We start by subtracting 211 from both sides. (5x=P-211) Next we divide by 5. [x=(P-211)/5] Therefore, [x=(P-211)/5], assuming P = perimeter

Chase makes 2 gallons of soup for a dinner party. He serves 10 cups of soup to his guests. How many cups of soup will be have left over?

Answers

2 gallons equal 32 cups

32-10= 22

22 cups will be left
He will have 22 cups left. 2 gallons = 32 cups minus 10.

Factor the polynomial.

4x7+32x+5-24x^4

A. 4x^4(x^3+8x-6)
B. 2x^4(2x^3+16x-12)
C. 2x^4(x3+8x-6)
D. 4x^4(2x^3+16x-12)

Answers

The coefficients here are 4, 32, and -24. The GCF of these values is 4.
As for the variables, the values are x^7, x^5, and x^4. The GCF is x^4
To factor this expression take out 4x^4 from each of the terms.
Coefficients should be divided by four and exponents should be subtracted by 4 since they have the same base (x).
The answer here is A.

the answer to number 3

Answers

3x^2 + 3x - 6

First factor out the 3 in front of x^2:

3( x^2 + x - 2 )

Now we can look at the inside of the brackets, and find two numbers that, when multiplied, give -2, and when added, give +1. You will usually find them if you create a list of factors of the constant term (-2 in this case).

2 * -1 = -2
2 + -1 = 1

So the factorization becomes 3(x+2)(x-1)

Sherita’s club is selling grapefruit to raise money. For every box they sell, they get $1.35 profit. They have sold 84 boxes already. How many more boxes must they sell to raise 270 dollars

Answers

1.35*84=113.4 dollars
to get to 270 they need 270-113.4=156.6
156.6/1.35=116 boxes

How to factor 2p^4+9p^3-18p^2

Answers

Factor out gcf

p^2(2p^2+9p-18)

Apply slip and slide to inner trinomial

p^2+9p-36
(p+12)(p-3)
(p+12/2)(p-3/2)
(p+6)(2p-3)

Final answer:
p^2(2p-3)(p+6)

Tim enlarged a picture with a width of 5.5 inches and a length of 8 inches by a scale factor of 3. What are the dimensions of the enlargement?

a. width: 24 in.; length: 16.5 in.b. width: 16.5 in.; length: 24 in.c. width: 14.5 in.; length: 22 in.d. width: 19.5 in.; length: 25 in.

Answers

It is B. 8*3 = 24. 5.5*3 = 16.5

Answer:

Option b. width = 16.5 inches and length = 24 inches

Step-by-step explanation:

Tim enlarged a picture wit a width of 5.5 inches and a length of 8 inches.

He enlarged the picture by a scale factor of 3.

We have to find the new dimensions of the picture.

Since, New dimension of the picture = Scale factor × dimensions before enlargement

So new width = 3×5.5 = 16.5 inches

new length = 3×8 = 24 inches

Therefore, option b. is the answer.

What numbesr can be multiplied and added to equal the same number?

Answers

It would be 6. for an example (3+2+1=6 | 3*2*1=6) 

2

2x2 = 4

2+2 =4


0

0x0=0

0+0=0


A gas station is 12 kilometers away. How far is the gas station in miles? Use the following conversion: 1 mile is 1.6 kilometers.

Answers

The gas station is 7.5 miles away
12/1.6=7.5 mies just divide the conversion

According to the text, which is not a main component of drawing?

Perspective
Vanishing point
Horizon

Answers

I think it's horizon

angle ABD and angle DBC are supplementary. Find the value of x.

A. 6
B. 8
C. 4
D. 10

Answers

Angle ABD and angle DBC are supplementary
so
14x+8x+4 = 180
22x = 176
x = 8

answer B. 8
The answer is B because if you substitute 8 to x and add them all up you will get 180 degrees.

12.5% of what is 130?

Answers

16.25 is 12.5% of 130
16.25 :/:/(;-;-((--((-(-(-(-(-(-(-(-(-((/)/)/

Find the domain of the following piecewise function

Answers

domain is the numbers you can use

see, we have -1≤x<0 and 0≤x<5

so if it is in the range of -1 to 5, then it is in the function

it includes -1 but not 5 because we gots -1≤x but x<5

remember, in interval notation, [] is includes and () is not include
or
[] is ≤≥ and () is <>

so
[-1,5)

what is the value of the fourth term in a geometric sequence for which a1=15 and r=1/3
express your answer as a fraction

Answers

nth term = a1r^(n-1)

4th term =  15* (1/3)^3  =  15/27 =  5/9

Answer:  The required fourth term in the given geometric sequence is [tex]\dfrac{5}{9}.[/tex]  

Step-by-step explanation:  We are given to find the fourth term of a geometric sequence with the following first term and common ratio :

[tex]a=15,~~r=\dfrac{1}{3}.[/tex]

We know that

the nth term of a geometric sequence with first term a and common ratio r is given by

[tex]a_n=ar^{n-1}.[/tex]

Therefore, the forth term of the given geometric sequence is

[tex]a_4=ar^{4-1}=ar^3=15\times\left(\dfrac{1}{3}\right)^3=15\times\dfrac{1}{27}=\dfrac{5}{9}.[/tex]

Thus, the required fourth term in the given geometric sequence is [tex]\dfrac{5}{9}.[/tex]  

Perform Gauss-Jordan elimination on the augmented matrix shown.
Could someone please help me figure this out.

Answers

Hi! Let me help you!
Basing from the visual you provided, it can be observed that item A, item B, item C, and item D have already undergone "row operations". Therefore, to complete Gauss-Jordan elimination, all we need to do is to transform row echelon A, row  echelon B, row echelon C, and  row echelon D such that each element they have separately become roots of their respective linear equations.

For A:
x = 2; y = (not sure if it is -1 or -7, the visual is too blurry for me to see)

For B:
x = ; y = -1

For C:
x = 0; y = 0

For D:
x = -(26/3); y = -7

I hope this helped you!

For an angle θ with the point (–20, –21) on its terminating side, what is the value of cosine?

Answers

The fact that the terminating side has 2 negative values tells us that it lies in Q3, where both x and y are negative. The cosine of an angle is the side adjacent/hypotenuse. But we don't have the value for the hypotenuse, so we have to find it using the Pythagorean Theorem. c^2 = -20^2 + -21^2  which gives us a c value of 29. Now that we have the hypotenuse length, we can set up the angle's cosine value: cos theta = -20/29.

Answer:

-20/29

Step-by-step explanation:

How could you find the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,-12)

Answers

I think it should be -6, because line segment equals 12 and it's midpoint will be 12/2 = 6, so you must go down by 6 from (0,0) point on y axis which is 0-6=-6

So the point will be (0, -6) 

A circle of radius 1 centered at (4, 0) is rotated about the y-axis.

1) Draw a picture of the three-dimensional shape that is produced when the circle is rotated about the y-axis.

2) In two or more complete sentences, describe the three-dimensional shape.

please help and thank you.

Answers

Answer to part 1) is shown in the diagram below

Answer to part 2)

By rotating a circle on the y-axis we obtain a solid shape called TORUS. This shape is like a pipe with its two ends joined. In older literature, it is called 'anchor ring' as its shape does look like a ring.

If h(x)=6-x, what is the value of (h o h)(10)

Answers

[tex](h \circ h)(x)=6-(6-x)=6-6+x=x\\\\ (h \circ h)(10)=10[/tex]

eight times the sum of a and b

Answers

Final answer:

The question concerns a basic algebraic expression 'eight times the sum of a and b' which is represented as 8(a + b). The expression emphasizes the operation order: sum first, then multiply, which results in a value eight times greater than the original sum.

Explanation:

The question 'eight times the sum of a and b' is a mathematical expression that can be represented as 8(a + b). This expression means you first add the numbers 'a' and 'b' and then multiply their sum by eight. The result you get after the multiplication will be eight times greater than the original sum of 'a' and 'b'. For instance, if 'a' is 2 and 'b' is 3, their sum is 5, and when this is multiplied by eight, it becomes 40, which is the desired expression's value.

To understand this concept further, we can refer to the exponentiation rule mentioned, which states that (xa)b = xa.b. Although this is a different type of operation—exponentiation—it demonstrates a similar principle of first performing the operation inside the parentheses and then applying the outside operation.

Finally, when performing algebraic operations, it's essential to remember that whatever you do to one side of the equation, you should do to the other side to maintain balance. This is a fundamental principle in algebra that helps to solve equations, such as the example provided showing how to isolate 'a' by subtracting 'x' from both sides of the equation a b.

If a train travels one mile (5,280 feet) while climbing a hill at an angle of five degrees, approximately how many vertical feet has the train climbed?

Answers

to calculate the vertical height multiply the hypotenuse ( 5280) by the sin of the angle (5)

5280 x sin(5) = 460.1823

round off to 460 feet

WILL GIVE BRAINLIST!!!!!!!!!!!!!! A group of students were surveyed to find out if they like working as a camp counselor and/or as a lifeguard during summer break. The results of the survey are shown below:

32 students like working as a camp counselor
8 students like working as a camp counselor but do not like working as a lifeguard
29 students like working as a lifeguard
8 students do not like working as a lifeguard or a camp counselor
Four students created the tables below to represent the data. CC represents camp counselor and LG represents lifeguard.

Which student's table is correct?

Answers

Answer:

I think Table C - Corralle

Answer:

c

Step-by-step explanation:

14÷420 long division

Answers

14➗ 420. 14 goes into 42 , 3 times subtract bring down the zero and your answer is 30

Find the simple interest paid for 8 years on a $850 loan at 7.5% per year.

Answers

The formula is
I=pre
I interest paid?
P principle 850
R interest rate 0.075
T time 8 years
I=850×0.075×8
I=510

The simple interest paid for 8 years on a $850 loan at a 7.5% annual interest rate is $510.

To calculate simple interest, you can use the formula:

Simple Interest (SI) = Principal (P) × Rate (R) × Time (T)

Where:

- Principal (P) is the initial amount of the loan.

- Rate (R) is the annual interest rate (in decimal form).

- Time (T) is the number of years.

In this case:

- Principal (P) = $850

- Rate (R) = 7.5% = 0.075 (decimal form)

- Time (T) = 8 years

Now, plug these values into the formula:

SI = $850 × 0.075 × 8

SI = $510

The simple interest paid for 8 years on a $850 loan at a 7.5% annual interest rate is $510.

For more such questions on Simple Interest :

https://brainly.com/question/30978175

#SPJ2

Find the probability of drawing a king from a standard deck of cards and then drawing a queen after the first card is replaced in the deck. None of the above 1/26 1/13 1/2704 2/13

Answers

52 cards in a deck

 4 kings

 4 queens

 King = 4/52 reduces to 1/13

Queen = 4/52 reduces to 1/13

1/13 *1/13 = 1/169

 so none of the above

With replacement:

P(K,Q)=(1/13)(1/13)

P(K,Q)=1/169

So none of the above.

Find the lateral area for the prism. L.A. =
Find the total area for the prism. T.A. =

Answers

The lateral area of the prism is given by:
LA=[area of the two triangles]+[area of the lateral rectangles]
hypotenuse of the triangle will be given by Pythagorean:
c^2=a^2+b^2
c^2=6^2+4^2
c^2=52
c=sqrt52
c=7.211'
thus the lateral area will be:
L.A=2[1/2*4*6]+[6*8]+[8*7.211]
L.A=24+48+57.69
L.A=129.69 in^2

The total are will be given by:
T.A=L.A+base area
base area=length*width
=4*8
=32 in^2
thus;
T.A=32+129.69
T.A=161.69 in^2

Answer:

Part 1) [tex]LA=(80+16\sqrt{13})\ in^{2}[/tex]

Part 2) [tex]TA=(104+16\sqrt{13})\ in^{2}[/tex]

Step-by-step explanation:

Part 1) Find the lateral area of the prism

we know that

The lateral area of the prism is equal to

[tex]LA=Ph[/tex]

where

P is the perimeter of the base

h is the height of the prism

Applying the Pythagoras Theorem

Find the hypotenuse of the triangle

[tex]c^{2}=4^{2}+6^{2}\\ \\c^{2}=52\\ \\c=2\sqrt{13}\ in[/tex]

Find the perimeter of triangle

[tex]P=4+6+2\sqrt{13}=(10+2\sqrt{13})\ in[/tex]

Find the lateral area

[tex]LA=Ph[/tex]

we have

[tex]P=(10+2\sqrt{13})\ in[/tex]

[tex]h=8\ in[/tex]

substitutes

[tex]LA=(10+2\sqrt{13})*8=(80+16\sqrt{13})\ in^{2}[/tex]

Part 2) Find the total area of the prism

we know that

The total area of the prism is equal to

[tex]TA=LA+2B[/tex]

where

LA is the lateral area of the prism

B is the area of the base of the prism

Find the area of the base B

The area of the base is equal to the area of the triangle

[tex]B=\frac{1}{2}bh[/tex]

substitute

[tex]B=\frac{1}{2}(6)(4)=12\ in^{2}[/tex]

Find the total area of the prism

[tex]TA=LA+2B[/tex]

we have

[tex]B=12\ in^{2}[/tex]

[tex]LA=(80+16\sqrt{13})\ in^{2}[/tex]

substitute

[tex]TA=(80+16\sqrt{13})+2(12)=(104+16\sqrt{13})\ in^{2}[/tex]




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