Use the properties of geometric series to find the sum of the given series. For what value of the variable does the series converge to this sum? 1+z/6+z^2/36+z^3/216+⋯ Enclose numerators, and denominators in parentheses. For example, (a−b)/(1+n). The sum of the given series is:

Answer:

b. 1,200 + 20 + 14

Step-by-step explanation:

617

x

2

______

1234

______

1234 aka. 1,200 + 20 + 14

The robot moves in the positive direction along a straight line so that after t minutes its distance is s=6t^4 feet from the origin. (a) Find the average velocity of the robot over intervals 2, 4. We have the following data: Initial time, t₁ = 2 min.

Final time, t₂ = 4 min.The distance from the origin is given by s = 6t^4Therefore, s₁ = s(2) = 6(2^4) = 6(16) = 96 feet s₂ = s(4) = 6(4^4) = 6(256) = 1536 feet

We can find the average velocity of the robot over the interval 2, 4 as follows: Average velocity = (s₂ - s₁) / (t₂ - t₁)Average velocity = (1536 - 96) / (4 - 2)Average velocity = 1440 / 2Average velocity = 720 feet per minute(b) Find the instantaneous velocity at t=2.To find the instantaneous velocity at t = 2 min, we need to take the derivative of the distance function with respect to time. We have the distance function as:s = 6t^4 Taking derivative of s with respect to t gives the velocity function:v = ds / dt Therefore,v = 24t³At t = 2, the instantaneous velocity is:v(2) = 24(2)³v(2) = 24(8)v(2) = 192 feet per minute Therefore, the instantaneous velocity at t = 2 min is 192 feet per minute.

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Substituting the value of y = x - 3 we get:-   x - 3 = x²+2x-4

What is Algebra?

One of the many branches of mathematics is algebra. Algebra, which is a common thread throughout practically all of mathematics, is broadly defined as the study of mathematical symbols and the rules for manipulating these symbols in formulas.

In the bottom equation, we need to determine the outcome of substituting for y.

Here is equation (1): y = x - 3

Here is equation (2): y=x²+2x-4

Add the y value from equation (1) to equation (2) in such a way that:

x - 3 = x²+2x-4

The equation above demonstrates that option (4) is the same as the equation above.

Hence, The correct option is D

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The way that the gallup can adjust their sample size is; to multiply it by 4

The formula for margin of error is;

E = zs/√n

where;

s is standard deviation

n is sample size

Thus;

Since we want to cut margin of error in half from 3 percentage points to 1.5 percentage points, since the standard deviation and z-score will remain the same, the way to do it will be to increase the sample size by multiplying it by 4.

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Answer:

x= 2.4

Step-by-step explanation:

Answer:

2.4 or 12/5

Step-by-step explanation:

1.2/0.96=3/x

convert the decimal into a fraction---- 6/5 /0.96= 3/x

Simplify the complex fraction-----5/4 = 3/x

simplify the equation using cross-mulitpication----- 5 x = 12

divide both sides of the equation by 5--- x= 12/5

Check if the solution is in the defined range

Alternitive form:

x=2 2/5, x =2.4

Answer:

\( \sqrt{i} \)

Step-by-step explanation:

i = -1

This is a complex number

in here we take -1 = i

Two alternators are being connected to supply power to a load. After the connection is made, in order to make the added alternator share the load, the input torque to the alternator being added must be increased.

Tο supply pοwer tο a lοad using multiple alternatοrs, the input tοrque tο the added alternatοr shοuld be adjusted tο match the pοwer demand οf the lοad. The added alternatοr shοuld be synchrοnized with the existing alternatοrs and cοnnected in parallel tο share the lοad.

The input tοrque tο the added alternatοr will depend οn the pοwer required by the lοad and the characteristics οf the alternatοrs, such as their pοwer ratings and efficiency. By prοperly adjusting the input tοrque οf the added alternatοr, it can cοntribute tο supplying pοwer tο the lοad alοng with the existing alternatοrs.

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The equivalent fraction for the decimal number 0.66 (repeating) is 2/3.

Let x = 0.666...

Then, 10x = 6.666...

Subtracting the left-hand side from the right-hand side, we get:

10x - x = 6.666... - 0.666...

Simplifying both sides, we have:

9x = 6

Dividing both sides by 9, we get:

x = 6/9

Simplifying the fraction, we get:

x = 2/3

Therefore, an equivalent fraction for the decimal number 0.66 (repeating) is 2/3.

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Answer:

+5

Step-by-step explanation:

Imagine (or draw) a right triangle with acute angles at the vertices (1, 1) and (5, 4).  The horizontal leg of this triangle has length (5 - 1), or 4, and the vertical leg has length (4 - 1), or 3.  Applying the Pythagorean Theorem:

(distance between these two points) = sqrt(3^2 + 4^2) = +sqrt(25) = +5

The graph of 3x-2y≤6 is the third graph, for 3x-2y<6 is the first graph, for 3x-2y>6 is the fourth graph and for 3x-2y≥6 is the second graph. The solution has been obtained using concept of linear inequality.

What is linear inequality?

A linear inequality is one that would produce a linear equation if the equals relation were used instead of the inequality. When multiplying or dividing both sides by a negative number in order to solve the inequality, the direction of the inequality is reversed. The entire set of solutions to an inequality is known as the solution set.

We are given for graphs, of which two graphs are dotted and two are simple straight line graphs.

The dotted graphs are drawn for the inequalities having < or >

Whereas the simple straight line graphs are drawn for the inequalities having ≤ or ≥.

Now, to notice the shaded pattern, we will see whether the equations are true for (0,0) or not

1. 3x-2y≤6

⇒ 0≤6

So, the equation is true for the point.

Hence, the third graph represents this equation.

2. 3x-2y<6

⇒ 0<6

So, the equation is true for the point.

Hence, the first graph represents this equation.

3. 3x-2y>6

⇒ 0>6

So, the equation is false for the point.

Hence, the fourth graph represents this equation.

4. 3x-2y≥6

⇒ 0≥6

So, the equation is false for the point.

Hence, the second graph represents this equation.

Hence, the graphs are matched with the inequalities.

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Since, there are multiple questions so, the question answered above is attached below.

Answer:

The speed of the plane in still air was 90 mph

The speed of the wind was 30 mph

Step-by-step explanation:

Your first step would be to make two equations so you can cancel them out and find what each variable is. So, I divided 1560 by 13 and got 120 mph which represents the plane's speed plus the wind's speed. Next, to find the speed of the trip to go back, I divided 1560 by 26 which got me to 60. This represents the plane's speed minus the wind's speed to find the way back. Once, you have these equations, line them up and use either method of elimination or substitution. I used elimination because you could see the variable "w" cancelling out very easily. Solve the equation and substitute the new value of the variable to get the other value.

p+w=120

p-w=60

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

2p=180

p=90

p+w=120--> 90+w=120--> w=30

W: 30 mph

P: 90 mph

I hope this helps!!

Answer:

The original price drops by 15%.

Step-by-step explanation:

What was the drop in price?  To answer that, subtract $21.25 from $25, which results in $3.75.

Now compare $3.75 to the original price, $25, through division, as follows:

   $3.75

-------------- * 100% = 15%

     $25

The original price drops by 15%.

The power function y = √(9x^5) can be written as\(y = kx^p\) as \(y = 3x^(5/2)\)

A power function is a single-term function where the real number, the coefficient, and the variable are each raised to a specified real number. To write the given power function in the form of \(y = kx^p\) it is required to   simplify the given equation,

\(y = √(9x⁵)\)

\(= √(9 × x⁴ × x)\)

\(= √(9 × ( x² )² × x)\)

\(= 3 × x² ×√x\)

\(= 3 × x² × x^(1/2)\)

\(= 3 × x^ (5/2)\)

The above equation can be written in the form \(y = kx^p\)where;

k = 3 and p = 5/2

Therefore, the power function of  y = √(9x^5) can written as y = 3x^(5/2)

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Answer:

rational number

Step-by-step explanation:

a whole number is a non-negative integer, a natural number is a positive integer, an integer is any number that doesn't have decimal places and isn't a fraction. 7/8 isn't an integer so it isn't a whole or natural number. rational numbers can be written as fractions or are terminating decimals. 7/8 is a fraction so 7/8 is a rational number

Step-by-step explanation:

What is the electric potential at the centre of square of side 1 m ? the changes 1 x 10^-8c , - 2 x 10^-8 c, 3 x 10^-8 c and 2 x 10^-8 c are placed at the corners

Answer:

Step-by-step explanation:

What's option 3?

Anyway, that's not the answer

The answer is a vertical line. x = - 3

Every y  on that line is a possible answer for y. So x has an infinite number of y values associated with it.

The growth in x years for the gym will be represented as \(y = 840(1.25)^x\).

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

City Fitness, a popular gym currently has 840 members. Based on its membership statistics since opening a few years ago, City Fitness expects to grow its membership by a factor of 1/4.

The expression can be written as,

\(y = 840(1.25)^x\).

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The 95% confidence interval for the population mean SAT score is given as follows:

(1340, 1460).

The confidence interval is calculated using the t-distribution, as the standard deviation for the population is not known, only for the sample.

The bounds are obtained according to the equation defined as follows:

\(\overline{x} \pm t\frac{s}{\sqrt{n}}\)

In which the variables of the equation are presented as follows:

In the context of this problem, the values of these parameters are given as follows:

\(\overline{x} = 1400, n = 64, s = 240\)

The critical value, using a t-distribution calculator, for a two-tailed 95% confidence interval, with 64 - 1 = 63 df, is t = 1.998.

Then the lower bound of the interval is of:

1400 - 1.998 x 240/sqrt(64) = 1340.

The upper bound of the interval is of:

1400 + 1.998 x 240/sqrt(64) = 1460.

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To answer the questions about group G with order 77: (1) Show that there are normal subgroups H and K of the group with order 7 and 11, respectively.

Since the order of G is 77, by the Sylow theorems, there exist Sylow 7-subgroups and Sylow 11-subgroups in G.

Let H be a Sylow 7-subgroup of G and K be a Sylow 11-subgroup of G. Since Sylow subgroups are conjugate to each other, H and K are both normal subgroups of G.

(2) Show that HK={hk|h∈H, k∈K} is an Abelian subgroup of group G.

Since H and K are normal subgroups of G, we have that HK is a subgroup of G. To show that HK is an Abelian subgroup, we need to prove that for any elements hk and h'k' in HK, their product is commutative.

Let hk and h'k' be arbitrary elements in HK. Since H and K are normal subgroups, we have that h'khk' = kh'h. Thus, the product hk h'k' is equal to kh'h, which implies that HK is an Abelian subgroup.

(3) Show that HK=G.

To show that HK=G, we need to prove that every element g in G can be expressed as a product hk, where h∈H and k∈K.

Since H and K are normal subgroups of G, their intersection H∩K is also a normal subgroup of G. By Lagrange's theorem, the order of H∩K divides both the order of H (which is 7) and the order of K (which is 11). Since 7 and 11 are coprime, the only possible order for the intersection is 1.

Thus, H∩K={e}, where e is the identity element of G. This implies that every element g in G can be uniquely expressed as g = hk, where h∈H and k∈K. Therefore, HK=G.

(4) Show that G is a cyclic group.

Since HK=G, and HK is an Abelian subgroup, we have that G is an Abelian group. Every Abelian group of prime order is cyclic. Since the order of G is 77, which is not prime, G cannot be cyclic.

Therefore, G is not a cyclic group.

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To answer the questions about group G with order 77: (1) Show that there are normal subgroups H and K of the group with order 7 and 11, respectively.

Since the order of G is 77, by the Sylow theorems, there exist Sylow 7-subgroups and Sylow 11-subgroups in G.

Let H be a Sylow 7-subgroup of G and K be a Sylow 11-subgroup of G. Since Sylow subgroups are conjugate to each other, H and K are both normal subgroups of G.

(2) Show that HK={hk|h∈H, k∈K} is an Abelian subgroup of group G.

Since H and K are normal subgroups of G, we have that HK is a subgroup of G. To show that HK is an Abelian subgroup, we need to prove that for any elements hk and h'k' in HK, their product is commutative.

Let hk and h'k' be arbitrary elements in HK. Since H and K are normal subgroups, we have that h'khk' = kh'h. Thus, the product hk h'k' is equal to kh'h, which implies that HK is an Abelian subgroup.

(3) Show that HK=G.

To show that HK=G, we need to prove that every element g in G can be expressed as a product hk, where h∈H and k∈K.

Since H and K are normal subgroups of G, their intersection H∩K is also a normal subgroup of G. By Lagrange's theorem, the order of H∩K divides both the order of H (which is 7) and the order of K (which is 11). Since 7 and 11 are coprime, the only possible order for the intersection is 1.

Thus, H∩K={e}, where e is the identity element of G. This implies that every element g in G can be uniquely expressed as g = hk, where h∈H and k∈K. Therefore, HK=G.

(4) Show that G is a cyclic group.

Since HK=G, and HK is an Abelian subgroup, we have that G is an Abelian group. Every Abelian group of prime order is cyclic. Since the order of G is 77, which is not prime, G cannot be cyclic.

Therefore, G is not a cyclic group.

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Answer:

C

Step-by-step explanation:

between three and four

Answer:

51 cm

Step-by-step explanation:

I think that because the

Just ans of 1 part cause I am lazy ~_~

Part 1 :

Area of whole figure:-

Answer:

a) $26

b) 24.07%

c) $94.3

Step-by-step explanation:

Given that:

Before tax:

Normal price of dress = $62

Discounted price of dress = $48

Normal price of a pair of shoes = $46

Discounted price of a pair of shoes = $34

a) Before the sales tax, total savings = ($64 - $48) + ($46 - $34) = $26

b) Total percentage discount on total sales.

Total bill for original price = $62 + $46 = $108

Percentage discount can be found by the formula:

\(\dfrac{\text{Total discount}}{\text{Total bill for original price}}\times 100\\\Rightarrow \dfrac{26}{108} \times 100 = 24.07\%\)

c) Total amount paid if the sales tax is 15%.

Amount paid with tax = Amount after discount + 15% of amount after discount

Amount after discount = $48 + $34 = $82

15% of $82 = $18.29

Amount paid with sales tax = $82 + $12.3 = $94.3

Answer:

1 in 5

Step-by-step explanation:

the answer is 120 just do 10*12

Answer:

The answer is 60.

Step-by-step explanation:

We are doing a triangle not a square, so divide 10*12 in half. What do you get? 60. That's your answer.

The equation log(8x+2) = 3 can be rewritten in exponential form as 10^3 = 8x+2. Simplifying, we get:

1000 = 8x + 2

998 = 8x

x = 124.75

Therefore, the solution to the equation is x = 124.75, which is an approximation to 3-decimal places.

To verify the solution, we can substitute x = 124.75 back into the original equation and check if both sides are equal:

log(8(124.75) + 2) = log(1000)

log(1000) = 3

Since both sides are equal, we can conclude that x = 124.75 is indeed the solution to the equation.

Note that it is important to check the solution obtained from solving the equation as sometimes the solution may not be valid due to the domain of the function involved in the equation.

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The number of miles in 4 weeks is 20 mules

From the question, we have the following parameters that can be used in our computation:

Stone runs 1 mile five times a week

This means that the rate is

Rate = 1 mile * 5 per week

So, we have

Rate = 5 miles per week

For 4 weeks, we have

Distance = 5 * 4

Evaluate

Distance = 20 miles

Hence, the distance is 20 miles

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Answer:

The amount that the movie-streaming service charges Justin's bank account each month is $13. To find out how much Justin will be charged in one year, we need to multiply this monthly charge by the number of months in a year (12):

Total charges in one year = Monthly charge x Number of months in a year

Total charges in one year = $13 x 12

Total charges in one year = $156

Therefore, the movie-streaming service will charge Justin's bank account a total of $156 in one year.