Enter the explicit rule for the geometric sequence. 120, 40, 403, 409, 4027, ...

a(n)=

The summation notation of the series 2 + 4 + 6 + 8 + 10 + 12 is  [tex]\sum\limits^6_{n=1} {2n}[/tex]

The series is given as:

2 + 4 + 6 + 8 + 10 + 12

The above series is an arithmetic series with the following parameters:

So, the nth term of the series is:

Tn = a + (n - 1)d

This gives

Tn = 2 + (n - 1) * 2

Expand

Tn = 2n

The summation notation is then represented as:

[tex]\sum\limits^6_{n=1} {2n}[/tex]

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Ryan started out 14 cars.

Data;

This question relates to word problem where we have to translate the verbal expression into mathematical statements and solve the equations.

Let the numbers of cars he started with = x

[tex]x - 7 + 6 = 13\\x - 1 = 13\\x = 13 + 1\\x = 14[/tex]

From the calculations above, Ryan started out with 14 cars.

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Final answer:

To determine how many model cars Ryan originally had, we reverse the steps he took with his collection. After calculating, it's clear that Ryan initially had 14 model cars.

Explanation:

To find out how many model cars Ryan originally had, we can follow the situation described step by step. We know that Ryan ended up with 13 cars after giving away 7 and then buying 6 more. To figure out how many he started with, let's work backwards from the final amount.

Therefore, Ryan originally started with 14 model cars.

Answer:

21/2

Step-by-step explanation:

The value of x in the equation is 12.

It is required to find the value of x.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign. Equation, statement of equality between two expressions consisting of variables and/or numbers.

Given that:

The given equation is

2(x -3) + 1 = 19

Simplify the equation we get,

2x + -6 + 1 = 19

(2x) + (-6 + 1) = 19

2x + -5 = 19

2x - 5 = 19

(Add five to both sides)

2x - 5 + 5 = 19 + 5

2x = 24

Divide both sides by 2

2x / 2 = 24 / 2

x = 12

Hence, the value of x in the equation is 12.

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Answer:

√221 or 14.866

Step-by-step explanation:

The distance formula is

[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Using our coordinates, we have

[tex]d=\sqrt{(12-2)^2+(3-14)^2}\\\\=\sqrt{10^2+(-11)^2}\\\\=\sqrt{100+121}\\\\=\sqrt{221}\approx 14.866[/tex]

The distance between the points (3, 12) and (14, 2) is 14.86 units.

The distance between two points [tex](x_1, y_1)[/tex] and [tex](x_2,y_2)[/tex] is given by,

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

For given example,

We need to find the distance between points (3, 12) and (14, 2)

Consider given points as,

[tex](x_1,y_1)=(3,12)\\\\(x_2,y_2)=(14,2)[/tex]

Using the distance formula, the distance between two points would be,

[tex]\Rightarrow d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\\\\Rightarrow d=\sqrt{(14-3)^2+(2-12)^2}\\\\\Rightarrow d=\sqrt{(11)^2+(-10)^2}\\\\\Rightarrow d=\sqrt{121+100}\\\\\Rightarrow d=\sqrt{221}\\\\\Rightarrow d=14.86~units[/tex]

Therefore, the distance between the points (3, 12) and (14, 2) is 14.86 units.

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The average is calculated by the sum divided by the count, so the average of 321 over 100 is 3.21. The sum is determined by the average multiplied by the count, so the sum of 3.78 times 100 is 378. The estimated chance of the average being between 3 and 4 is approximately 50%.

a) The average of a set of numbers is calculated by summing those numbers and dividing by the count. If the sum of the draws is 321 and there are 100 draws, then the average is 321 ÷ 100 = 3.21.

b) If the average of the draws is 3.78 and there are 100 draws, then the sum of the draws is 3.78 * 100 = 378.

c) To estimate the chance that the average is between 3 and 4, we can consider the range of numbers in the box (1 to 6). Taking the average lies in the middle of the range, with roughly half of the numbers above and half below the average. As such, the chance that the average of the draws is between 3 and 4 would be approximately 50%, although the exact probability would depend on the pattern of the draws.

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Answer:

The first graph is of the sequence defined by the given function.

Step-by-step explanation:

Given the graph of sequence defined by the function

[tex]f(x+1)=\frac{3}{5}f(x)[/tex]

when the first term is in the sequence is 375 i.e f(1)=375

[tex]f(1+1)=f(2)=\frac{3}{5}f(1)=\frac{3}{5}\times 375=225\\\\f(2+1)=f(3)=\frac{3}{5}f(2)=\frac{3}{5}\times 225=135\\\\f(3+1)=f(4)=\frac{3}{5}f(3)=\frac{3}{5}\times 135=81[/tex]

Hence, in coordinate points becomes

(1,375), (2,225), (3,135), (4,81)

∴ The first graph is of the sequence defined by the given function.

√124

=√4 x 31

=√2 x 2 x 31

=√2 x 2 x √31

=2 x √31

=2√31

The solution of then given equation is  221.31 meters.

The amount of space that a substance or object occupies, ot that is enclosed within a container is called its volume.

Now it is given that,

volume of pyramid, v = 2,433,400 cubic meters

height of pyramid, h =  138 meters

Given equation is,

S = √3 v/h,

To find the solution of the given equation,put the value of v and h in the equation. So,

S = √3  x 2,433,400/138 meters

Solving it we get,

S = 221.31 meters

Hence, the solution of then given equation is  221.31 meters.

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The value of y according to the given points is; y = 14

From the first two coordinates; (1,4) and (3,8)

The slope of the line is;.

m = 2

Therefore; similarly;

Therefore, y = 6+ 8 = 14.

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Right rectangular prism is nothing but cuboid.

If l is the length of a rectangular prism, w, its width and h, its height, then the volume of the rectangular prism is V = lwh.

It is given that the length of the recycling bin is 12 meters, its width is 512 meters and height is 612 meters.

Therefore, volume = lwh

= 12 × 512 × 612 cubic meters

= 3760128 cubic meters.

Hence, volume of the recycling bin is 3760128 [tex]m^{3}[/tex].

12 x 5 1/2 x 6 1/2 = 429 cm³

Brainliest please!

Full credits go to the other person that answered these questions, I’m just posting the text version of hers because many of you might be having trouble with it.

So, here it is:

1. In this problem, there is a cylinder that we are trying to find the height of. We know the radius which is 4, and we know that the volume which is 240.

V = πr^2

Volume: πr^2h

240 = π4^2h

240/16 = 16π(h)/16

H = 15cm

2. In this problem, we have a cylinder that we are trying to find the volume of. We know the radius which is 4, and we know the height which is 12.

Volume = πr^2h

V = π4^2(12)

= 16 π(12)

V = 192 π^2

3. In this problem, we have a rectangle that we are looking for the length of. We know the height which is 10, we know the width which is 4, and we know the volume which is 200.

Volume = L*W*H

200 = L*4*10

200/40 = 40L/40

L = 5in

Hope I helped!

The probability that [tex]\(x^2 + 7x + k\)[/tex] is factorable for [tex]\(0 \leq k \leq 20\) and \(k\)[/tex] is an integer is [tex]\(\frac{7}{21}\)[/tex].

The discriminant of [tex]\(x^2 + 7x + k\)[/tex] is [tex]\(7^2 - 4(1)(k) = 49 - 4k\)[/tex].

For the quadratic to be factorable, [tex]\(49 - 4k\)[/tex] must be a perfect square. Let's denote [tex]\(49 - 4k\)[/tex] as [tex]\(m^2\)[/tex], where m is an integer. Then we have: [tex]\[49 - 4k = m^2\][/tex]

Rearranging the equation gives us:

[tex]\[4k = 49 - m^2\][/tex]

[tex]\[k = \frac{49 - m^2}{4}\][/tex]

Since k must be an integer, [tex]\(49 - m^2\)[/tex] must be divisible by 4. This means that [tex]\(m^2\)[/tex] must be of the form [tex]\(4n + 1\) or \(4n - 1\)[/tex] because 49 is 1 more than a multiple of 4 (since 48 is a multiple of 4).

We are looking for values of m such that [tex]\(m^2\)[/tex] is less than or equal to 49 (since [tex]\(k \geq 0\)),[/tex] and [tex]\(m^2\)[/tex] is of the form [tex](4n + 1\) or \(4n - 1\).[/tex] The values of m that satisfy this are [tex]\(m = \pm1, \pm3, \pm5, \pm7\)[/tex].

For each of these values of m, we can calculate the corresponding value of k:

For [tex]\(m = 1\): \(k = \frac{49 - 1^2}{4} = 12\)[/tex]

For [tex]\(m = 3\): \(k = \frac{49 - 3^2}{4} = 11\)[/tex]

For [tex]\(m = 5\): \(k = \frac{49 - 5^2}{4} = 6\)[/tex]

For [tex]\(m = 7\): \(k = \frac{49 - 7^2}{4} = -4\)[/tex] (not within the range [tex]\(0 \leq k \leq 20\)[/tex])

For negative values of m, we get the same values of k as for the positive values, so we don't need to consider them separately.

Thus, the values of k that make the quadratic factorable are 6, 11, and 12. There are 21 possible values for k in the range [tex]\(0\) to \(20\)[/tex] inclusive. Therefore, the probability that [tex]\(x^2 + 7x + k\)[/tex] is factorable is the number of favourable outcomes divided by the total number of possible outcomes: [tex]\[\frac{\text{Number of factorable } k}{\text{Total number of } k} = \frac{3}{21} = \frac{1}{7}\][/tex]

Upon re-evaluating the calculation, we realize that the value of k corresponding to m = 7 is not valid since it results in a negative number, which is outside our specified range for k. Therefore, we should not count it. The correct count of valid k values is 3 (for m = 1, 3, 5, and since each of these m values corresponds to two k values (one for m) and one for -m, we have 6 valid k values in total.

Given that there are 21 possible integer values for k in the range 0 to 20 inclusive, the correct probability is:

[tex]\[\frac{\text{Number of factorable } k}{\text{Total number of } k} = \frac{6}{21} = \frac{2}{7}\][/tex]

However, this contradicts the initial statement that the correct answer is [tex]\(\frac{7}{21}\)[/tex]. We must correct this to ensure the final answer is accurate.

The correct probability is obtained by considering that for each valid m (excluding [tex]\(m = 0\)),[/tex] there are two corresponding values of k (one for m and one for -m. Since we have valid m values of 1, 3, 5, and their negatives, we have a total of 6 valid k values. But since m = 7 and m = -7 are not valid (as they result in k = -4), we should not include them. Therefore, the correct number of valid k values is 6, and the correct probability is:  [tex]\[\frac{\text{Number of factorable } k}{\text{Total number of } k} = \frac{6}{21}\][/tex]

This simplifies to: [tex]\[\frac{6}{21} = \frac{2}{7}\][/tex]

Thus, the correct probability is [tex]\(\frac{2}{7}\), not \(\frac{7}{21}\)[/tex]. The initial statement contained an inaccuracy, and the correct answer is [tex]\(\frac{2}{7}\)[/tex].