Mathematics High School

Which geometric series converges?
A. 2+0.2+0.02+0.002+...
B. 2+4+8+16+...
C. 2-20+200-2000+...
D. 2+2+2+2+...

Answers

Answer 1

Answer:  The correct option is (A) 2 + 0.2 + 0.02 + 0.002 +  .  .  .

Step-by-step explanation:  We are given to select the correct geometric series that converges.

We know that

a geometric series converges if the modulus of its common ratio is less than 1.

Option (A) : 2 + 0.2 + 0.02 + 0.002 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

[tex]r=\dfrac{0.2}{2}=\dfrac{0.02}{0.2}=\dfrac{0.002}{0.02}=~.~.~.~=0.1\\\\\Rightarrow |r|=|0.1|=0.1<1[/tex]

So, this geometric series will converge.

Option (A) is correct.

Option (B) : 2 + 4 + 8 + 16 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

[tex]r=\dfrac{4}{2}=\dfrac{8}{4}=\dfrac{16}{8}=~.~.~.~=2\\\\\Rightarrow |r|=|2|=2>1.[/tex]

So, this geometric series will not converge.

Option (B) is incorrect.

Option (C) : 2 - 20 + 200 - 2000 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

[tex]r=\dfrac{-20}{2}=\dfrac{200}{-20}=\dfrac{-2000}{200}=~.~.~.~=-10\\\\\Rightarrow |r|=|-10|=10>1.[/tex]

So, this geometric series will not converge.

Option (C) is incorrect.

Option (D) : 2 +2 + 2 + 2 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

[tex]r=\dfrac{2}{2}=\dfrac{2}{2}=\dfrac{2}{2}=~.~.~.~=1\\\\\Rightarrow |r|=|1|=1.[/tex]

So, this geometric series will not converge.

Option (D) is incorrect.

Thus, the correct option is (A).

Answer 2

The geometric series that converges is 2+0.2+0.02+0.002+ ...:

Thus, option (A) is correct.

In a geometric series, the terms are multiplied by a constant ratio to obtain the next term.

If the absolute value of the common ratio is less than 1, the series converges.

A. 2 + 0.2 + 0.02 + 0.002 + ...:

In this series, the common ratio is 0.1 (each term is divided by 10).

Since the absolute value of the common ratio is less than 1, this series converges.

B. 2 + 4 + 8 + 16 + ...:

In this series, the common ratio is 2 (each term is multiplied by 2).

Since the common ratio is greater than 1, this series diverges.

C. 2 - 20 + 200 - 2000 + ...:

In this series, the terms alternate in sign, but the absolute value of the common ratio is 10.

Since the absolute value of the common ratio is greater than 1, this series diverges.

D. 2 + 2 + 2 + 2 + ...:

In this series, the common ratio is 1 (each term is the same).

Since the common ratio is equal to 1, this series neither converges nor diverges. It is a divergent series.

Therefore, the geometric series that converges is 2 + 0.2 + 0.02 + 0.002 + ...

Thus, option (A) is correct.

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Related Questions

Toby gets 78 votes, which is 52% of the total votes cast. How many students voted in Toby’s grade?

Answers

52% = 0.52

78/0.52 =150

 150 students voted


150 students voted in Toby's grade.

What are the vertical asymptotes of the function f(x) = the quantity of 2 x plus 8, all over x squared plus 5 x plus 6? x = −3 and x = −2 x = −3 and x = 2 x = 1 and x = −2 x = 1 and x = 2?

Answers

for this question the answer would be x=-3 and x=-2

What are the explicit equation and domain for a geometric sequence with a first term of 3 and a second term of −9?

Answers

u can find the common ratio by dividing the 2nd term by the first term
r = -9/3 = -3

an = a1 * r^(n - 1)
a1 = 1st term = 3
r = common difference = -3
now sub
an = 3 * -3^(n - 1) <== ur formula
domain : all integers where n > = 1 <==

Answer: [tex]\ a_{n}=3(-3)^{n-1}[/tex]


Step-by-step explanation:

Given: A geometric sequence with its first term [tex]a_1=a=3[/tex]

and second term [tex]a_2=-9[/tex]

We know that the common ratio in of a geometric sequence=[tex]\frac{a_{n}}{a_{n-1}}[/tex]

Thus, common ratio [tex]r=\frac{-9}{3}=-3[/tex]

We know that the explicit rule for geometric sequence is written as

[tex]a_{n}=ar^{n-1}\\\Rightarrow\ a_{n}=3(-3)^{n-1}..[\text{by substituting the values of 'a' and 'r' in it }][/tex]

Thus, the explicit rule for the given geometric sequence is [tex]\ a_{n}=3(-3)^{n-1}[/tex] for every n ,a natural number.

Which expression is equivalent to (cos x)(tan(–x))?

A. -sin x
B. sin x
C. -csc x
D. csc x

Answers

The tan(-x) is the same thing as -tan(x).  The tangent function is also the same thing as sin(x)/cos(x), right? So let's rewrite that tan in terms of sin and cos:
[tex][cos(x)][tan(-x)][/tex] is the same as [tex][cos(x)][ -\frac{sin(x)}{cos(x)}] [/tex]
We can now cancel out the cos(x), which leaves us only with -sin(x) remaining. So your answer is A.
the answer to this is A 

A model for a company's revenue is R=-15p^2+300p+12,000, where p is the price in dollars of the company's product. What prize will maximize revenue?

Answers

check the picture below.

[tex]\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{lccclll} R = &{{ -15}}p^2&{{ +300}}p&{{ +12,000}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)[/tex]

so the Revenue will be the highest at   [tex]\bf {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}[/tex]   and that will happen at the price of    [tex]\bf -\cfrac{{{ b}}}{2{{ a}}}[/tex]
In order to do this you have to complete the square to get the vertex of the equation. The vertex will tell you the max value or the min value of whatever it is we are looking for. Here, our x values are the prices of whatever we are selling and the y values are the revenue dollars. Completing the square looks like this for us:
[tex]R(p)=-15 p^{2}+300p+12,000[/tex]
Move the 12,000 over to the other side and set the equation equal to 0:
[tex]-15 p^{2}+300p=-12,000[/tex]
In order to complete the square, the leading coefficient on the squared term has to be a 1 and it's a -15, so we have to factor that out:
[tex]-15( p^{2}-20p)=-12,000 [/tex]
Now it's ready to complete the square on it. Do this by taking half the linear term, squaring it, and then adding it in. Our linear term is 20p.  Half of 20 is 10 and 10 squared is 100, so we will add in 100 on the left. Let's do this next step in two parts:
[tex]-15( p^{2}-20p+100)=-12,000 [/tex]
That's not quite complete yet because if we add in 100 inside the parethesis on the left we have to add in that same amount on the right. But on the left, we didn't just add in 100, we added in -15 TIMES 100 cuz we can't just forget about the -15 we factored out.  That looks like this then:
[tex]-15( p^{2}-20p+100)=-12,000-1,500 [/tex]
The whole reason for doing this is to create a perfect square binomial on the left which is what we have done. The perfect square binomial is this (and we are going to do the subtraction on the right at the same time):
[tex]-15(p-10)^{2}=-13,500[/tex]
If we move the right side over to the left we get this:
[tex]-15(p-10) ^{2}+13,500=0 [/tex]
which gives us a vertex of (10, 13,500).  That means that at a price of $10, the max revenue will be $13,500.  See how beautifully that works out!??


URGENT!!!!!!!!!!11!! Plz HElP

A school typically sells 500 yearbooks each year for $50 each. the economics class does a project and discovers that they can sell 100 more yearbooks for every $5 decrease in price.the revenue for the yearbook sales is equal to the number of yaerbooks sold times the price of the yearbook. let x represent the number of $5 decreases in price. if the expression that represents the revenue is written in the form r(x)=(500+ax)(50-bx). find the values of a and b.

Answers

The answer is a=100 and b=5

Line LJ is a diameter of circle K. If tangents to circle K are constructed through points L and J, what relationship would exist between the two tangents? Explain.

Answers

A tangent line to a circle is perpendicular to the radius drawn to the tangent point.

If tangents are constructed through points L and J, then the tangents are perpendicular to the diameter and parallel to each other.

How much water can be held by a cylindrical tank with a radius of 12 feet and a height of 30 feet?

Answers

13571.68 cube feet of water can be held

5⁄6 · n = 10 (solve for n)

Answers

You divide 5/6 from the equation.
When you divide 5/6 and 10 you get 12.
To check your answer multiply 5/6 and 12 and you will get 10.
N=12
[tex]\frac{5}{6}n=10\ | \div \frac{5}{6} \\\\ n= \frac{\not10^2}{1}\cdot \frac{6}{\not5^1}\\\\n=2 \cdot 6=12 \\\\ or \\\\ \frac{5}{6}n=10 \ | \times 6 \\\\ 5n=60 \ |\div 5 \\\\ n=12[/tex]

Sophie has a hard rubber ball whose circumference measures 13 inches. she wants to store it in a box. what is the number of cubic inches in the volume of the smallest cube- shaped box with integer dimensions that she can use?

Answers

Let r = radius of the hard rubber ball.

Because the circumference of the ball is 13 inches, therefore
2πr = 13
r = 13/(2π) = 2.069 in

In order to fit the ball into a cube-shaped box with integer dimensions, each side of the box should measure 3 inches (the next integer greater than 2.069).

The volume of the cube-shaped box is 3³ = 27 in³.

Answer: The volume is 27 in³.

Flying against the wind, a jet travels 4400mi in 8 hours. Flying with the wind, the same jet travels 5820mi in 6 hours. What is the rate of the jet in still air and what is the rate of the wind?

Answers

Let J = rate of jet in still airLet W = rate of wind Distance formula:  d = rt   /   r = d/t Flying against the wind the jet flies at a rate of  J - W = 1860 miles/3 hours = 620 miles per hourFlying with the wind the jet flies at a rate of J+W = 9180 miles/9 hours = 1020 miles per hour The average of these 2 rate is the speed of the jet in still air J = (620+1020)/2 = 820 miles per hour J - W = 620  

820 - W = 620
W = 820 - 620 = 200 miles per hour The jet in still air flies at a rate of 820 mph(miles per hour)The wind speed is 200 mph(miles per hour)

Explain what needs to happen to the inequality sign when dividing or multiplying by a negative number. a. nothing happens c. change the inequality sign to an equals sign b. flip the inequality sign d. the inequality needs to be graphed on a number line

Answers

flip the inequality sign 
so the answer is B

Answer:

(B) flip the inequality sign.

Step-by-step explanation:

If we consider an inequality such that [tex]-x\leq7[/tex], then if we multiply the inequality with a negative number such as [tex]-1[/tex], then the inequality becomes [tex]x\geq-7[/tex].

Also, if we divide the above inequality  [tex]-x\leq7[/tex], by a negative number that is  [tex]-1[/tex], then the inequality becomes [tex]x\geq-7[/tex].

Therefore, if we multiply or divide an inequality by a negative number, then it flips the inequality sign.

Hence, option (B) is correct.

What is the prime factorization of -96? tell me how do u get the answer

Answers

hello : 
 the prime factorization of -96 is :  - 3×2^5

Answer: -1 x 2^5 x 3

Step-by-step explanation: To find the prime factorization of -96, we need to first factor out -1, which gives us 1 x (-1) x 2 x 2 x 2 x 2 x 2 x 3. Then, we can rewrite -1 as -1^1, and combine the 2's to get 2^5. So the prime factorization of -96 is -1^1 x 2^5 x 3.

Remember, negative numbers can also have prime factorizations.

From a group of six people, two individuals are to be selected at random. how many possible selections are there

Answers

Final answer:

To find the number of possible selections when choosing two individuals from a group of six people at random, use the combinations formula C(6, 2) = 6! / (2!(6-2)!) which equals 15 possible selections.

Explanation:

From a group of six people, selecting two individuals at random involves calculating the number of combinations. The formula for combinations is C(n, k) = n! / (k!(n-k)!), where 'n' represents the total number of items and 'k' represents the number of items to choose.

In this scenario, with n being 6 (the total number of people) and k being 2 (the number of people to select), the calculation would be:

C(6, 2) = 6! / (2!(6-2)!) = (6 * 5) / (2 * 1) = 15

Therefore, there are 15 possible selections when choosing two individuals from a group of six people at random.

Accuracy is a measure of how close an answer is to the actual or expected value

Answers

That statement is true
this statement is accurate having 100 % accuracy .

:)

house blend coffee is 50% columbian beans and special blend coffee is 80% columbian beans. how much of each should be used to produce 100kg of a blend that is 68% columbian beans

Answers

Let us say that,

x = mass of 50% Columbian beans required

y = mass of 80% Columbian beans required

To solve this problem, we set up two mass balance equations:

Overall mass balance:   100 = x + y

                                    x = 100 – y                                ---> 1

Coffee mass balance:   0.68 (100) = 0.50 x + 0.80 y

                                    68 = 0.50 x + 0.80 y                   ---> 2

Combining equations 1 and 2:

68 = 0.50 (100 – y) + 0.80 y

68 = 50 – 0.50 y + 0.80 y

18 = 0.30 y

y = 60 kg

 

Calculating for x using equation 1:

x = 100 – y

x = 100 – 60

x = 40 kg

 

Answer:

40 kg of 50% Columbian beans required

60 kg of 80% Columbian beans required

If you guess an answer on two multiple choice questions with the options a, b, or c, what is the probability of you guessing the answer to both questions correctly?

Answers

first, you find the probability of getting each question right individually, then you multiply the two probabilities together.

(1/3) x (1/3) = 1/9

The stack on the left is made up of 15 pennies, and the stack on the right is also made up of 15 pennies. If the volume of the stack of pennies on the left is 360 mm3, what is the volume of the stack of pennies on the right in cubic millimeters? Use 3.14 for pi. (Hint: only enter numerals in the answer blank)

Answers

the volumes are the same.



Answer:

360 is the answer. Just type in 360 in the answer blank.

Step-by-step explanation:

In how many different ways can a president vice-president and secretary be elected from a class of 15 students

Answers

Answer: 2730

---------------------------------------------------------

Explanation:

A shortcut way is to start at 15 and count down til you reach three values. So that means you'll have 15, 14, 13 which multiplies out to...

15*14*13 = 2730

Or you can use the nPr formula, with n = 15 and r = 2, to get the same answer

nPr = (n!)/((n-r)!)
15P3 = (15!)/((15-3)!)
15P3 = (15!)/(12!)
15P3 = (15*14*13*12!)/(12!)
15P3 = 15*14*13
15P3 = 2730

It costs $35$35 per hour to rent a boat at the lake. You also need to pay a $25$25 fee for safety equipment. You have $200$200. For how long can you rent the boat?

Answers

5 hours because 35+25 divided by 200
                                                            

Indicate the equation of the given line in standard form. The line that contains the point Q( 1, -2) and is parallel to the line whose equation is y - 4 = 2/3 (x - 3)

Answers

the equation given has a slope of 2/3. A parallel line will have the same slope.

y = mx + b
slope(m) = 2/3
(1,-2)...x = 1 and y = -2
sub and find b, the y int
-2 = 2/3(1) + b
-2 = 2/3 + b
-2 - 2/3 = b
-6/3 - 2/3 = b
- 8/3 = b

equation is : y = 2/3x - 8/3...but we need it in standard form

y = 2/3x - 8/3
-2/3x + y = -8/3...multiply by -3
2x - 3y = 8 <== standard form

Find the area of a square with apothem 9 in. Round to the nearest whole number.

281 in2


305 in2


458 in2


324 in2

Answers

The apothem is the line from the center of the polygon (square) to the midpoint of a side.

So, if the apothem is 9in, the length of the side is 2 * 9in = 18 in.

And, the area of the square is (length of the side)^2 = (18 in)^2 = 324 in^2

Answer: 324 n^2

Answer:

The area of the square is 324 square inches.

Step-by-step explanation:

The apothem of the square is 9 inches.

The side of the square is twice the length of the apothem.

Hence, the side of the square is given by

[tex]a=2\times 9=18\text{ in}[/tex]

The area of a square is the given by

[tex]A=a^2\\A=18^2\\A=324\text{ in}^2[/tex]

Therefore, the area of the square is 324 square inches.

A comic-strip writer churns out a different number of comic strips each day. For 16 days, the writer logged the number of comic strips written each day (sorted low to high): {1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7}. What type of skew can be observed in this distribution?

positive skew

negative skew

zero skew

skew cannot be observed

Answers

One way to observe the skewness of a data set is to find the quartiles: Q₁, Q₂, Q₃ and then sketch the box plot

We have the data set already in ascending order, so finding the quartiles is quite straight forward. 

We have Q₁ = 2, Q₂ = 3, Q₃ = 5 (refer to the first picture below)

The box plot is given in the second picture and from this plot, we can see that the data tail slightly on the right, and this shows a positive skew.

One number is 6 less than another. their sum is 12 find the larger number

Answers

y=x-6

x+y=12

x+x-6 =12

2x-6=12

2x=18

x=18/2 =9

x=9

y =9-6 =3

9+3 =12

Larger number is 9


How do I solve this

Answers

[tex]\bf \cfrac{x+2}{x^2+6x-7}[/tex]    so, that function is "defined", ok, what values of "x" are not in the domain, namely, what values can "x" take on and not make the function "undefined", well,  you know, if we end up with a 0 at the denominator, like   [tex]\bf \cfrac{x+2}{0}[/tex]    then, we'd have an "undefined" expression...so... any values of "x" that make the denominator 0, are not really the ones we want, and thus they'd be excluded from the domain.


so, hmm which are those? let's check, let's set the denominator to 0, and solve for "x".

[tex]\bf x^2+6x-7=0\implies (x+7)(x-1)=0\implies x= \begin{cases} -7\\ 1 \end{cases} \\\\\\ \textit{let's check, } x=-7\quad \cfrac{(-7)+2}{(-7)^2+6(-7)-7}\implies \cfrac{-5}{49-42-7}\implies \cfrac{-5}{0} \\\\\\ x=1\quad \cfrac{(1)+2}{(1)^2+6(1)-7}\implies \cfrac{3}{1+6-7}\implies \cfrac{-3}{0}[/tex]

PLEASE CAN SOMEONE HELP!!!????

Use elimination to solve for x and y:

−2x−y=9

2x−9y=1

My Choices are....

a. (−4,−1)
b. (−1,−4)
c. (5,1)
d. (−1,−7)

Answers

add them 2 equations to eliminate x's

-2x-y=9
2x-9y=1 +
0x-10y=10

-10y=10
divide by -10
y=-1

sub back

-2x-y=9
-2x-(-1)=9
-2x+1=9
-2x=8
divide by -2
x=-4

y=-1

(x,y)
(-4,-1)


A

The values of x and y after solving both the equations by the elimination method are  -4 and -1. So option A is correct

What is an Equation ?

An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.

For example, 3x+2y=0.

Types of equation

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

Two linear equations

−2x − y = 9

2x − 9y = 1

by using elimination method,

−2x − y = 9

2x − 9y = 1  

0  - 10y  =  10

y  = -1

2x − 9× -1 = 1

2x   = -8

x  = -4

Hence, the values are, -4 and -1

To know more about Equation check:

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A ship travels 10 miles from Point A to Point B, makes a turn of 112, and travels 16 miles to Point C. If the ship travels directly from Point C back to Point A, how many miles will it travel on the last leg of the trip (from Point C to Point A)? Round your answer to the nearest tenth of a mile.

Answers

This is Pythagorean's Theorem, with one leg being 10 and the hypotenuse being 16.  Using those values in the theorem looks like this:
[tex] 10^{2} + b^{2} = 16^{2} [/tex]
and [tex]100+ b^{2} =256[/tex]
and [tex] b^{2} =256-100[/tex]
so [tex] b^{2} =156[/tex]
Take the square root of both sides to find that b = 12.5

Permutations!!
If 9 actors must sit together how many ways are there to seat 13 people around the table?

Answers

Final answer:

To calculate the number of ways to seat 13 people around a table with 9 actors sitting together, we treat the 9 actors as one unit and then arrange the five units around the table, resulting in (4! * 9!) different possible arrangements.

Explanation:

The question asks us to calculate the number of ways to seat 13 people around a table if 9 actors must sit together. This can be approached as a permutations problem in combinatorics.

Firstly, treat the 9 actors as one unit since they must sit together. With this in mind, we effectively have 5 units to arrange: the group of 9 actors and the remaining 4 individuals. As the seating arrangement is around a circular table, we can fix one person's seat and arrange the remaining units. As a result, there are (5-1)! ways to arrange these units since circular permutations eliminate the concept of a distinct 'starting' point that linear permutations have.

Now we need to consider the arrangements of the 9 actors within their group. Since their relative positions to each other matter, they can be permuted in 9! ways.

Therefore, the total number of seating arrangements would be the product of the two permutations: (5-1)! * 9!.

Calculating this gives us (4!) * 9! = (4*3*2*1) * (9*8*7*6*5*4*3*2*1) different possible arrangements.

Jake has proved that a function, f(x), is a geometric sequence. How did he prove that?
A He showed that an explicit formula could be created.
B He showed that a recursive formula could be created.
C He showed that f(n) ÷ f(n − 1) was a constant ratio.
D He showed that f(n) − f(n − 1) was a constant difference.

Answers

The defining characteristic of any geometric sequence is the common ratio, which is the constant found when dividing any term by the term preceding it.

C.  He showed that f(n)/(f(n-1) was a constant ratio.
The answer to your question is : C. He showed that f(n) ÷ f(n − 1) was a constant ratio.

Why does a bag of chips puff up in an airplane?

Answers

Because of the pressure differences.
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