Refer to the attachment and answer these questions.
( 1 ). Find the first counting number 'x' in group n.
( 2 ). Find the sum S of all counting numbers in group n.
Please also show your work too so others and I can understand. Thanks in advance!

Refer To The Attachment And Answer These Questions.( 1 ). Find The First Counting Number 'x' In Group

Answers

Answer 1

Answer:

[tex]\displaystyle \text{(1)}\\x=1+\sum_{k=1}^{n-1}2k\\\\\text{(2)}\\\displaystyle \frac{\displaystyle \sum_{k=1}^{n}2k\left(1+\displaystyle \sum_{k=1}^{n}2k \right)}{2}-\frac{\displaystyle \sum_{k=1}^{n-1}2k\left(1+\displaystyle \sum_{k=1}^{n-1}2k \right)}{2}[/tex]

Step-by-step explanation:

Notice that Group 1 starts with 1 and we keep counting up by 1.

Part 1:

The total number of numbers from Group 1 to Group [tex]n[/tex] can be found by:

[tex]\displaystyle \sum_{k=1}^{n}2k[/tex]

To find the first number in a certain group [tex]g[/tex], we'll find the total number of numbers from Group 1 to Group [tex]g-1[/tex], then add 1.

Here's why.

Let's say we want to find the first number of Group 3, which we can clearly see is 7. Take advantage of the fact that the sequence of numbers counts up by 1, starting from 1. We will use the formula above to to find the total number of numbers from Group 1 to Group 2, which is 6, then add 1 to achieve the number of 7.

Therefore, the first number in group [tex]n[/tex] must be:

[tex]\boxed{\displaystyle 1+\sum_{k=1}^{n-1}2k}[/tex]

Part 2:

Recall that the sum of the numbers from [tex]1[/tex] to positive integer [tex]n[/tex] inclusive is given by [tex]\displaystyle\frac{n(n+1)}{2}[/tex].

Therefore, if we find the total number of numbers from Group 1 to Group [tex]n[/tex], we can simply substitute that value into the formula above to find the total sum of the numbers from Group 1 to Group [tex]n[/tex].

To find the total sum of the numbers exclusively in Group [tex]n[/tex] simply subtract the sum of numbers from Group 1 to Group [tex]n-1[/tex].

Short example on how that works:

The sum of all numbers in Group 3 is equal to the sum of the numbers from Group 1 to Group 3 minus the sum of the numbers from Group 1 to Group 2.

We've already found the formula for the total number of numbers from Group 1 to Group [tex]n[/tex] in the previous part, so convert this concept into summation notation like such:

[tex]\boxed{\displaystyle \frac{\displaystyle \sum_{k=1}^{n}2k\left(1+\displaystyle \sum_{k=1}^{n}2k \right)}{2}-\frac{\displaystyle \sum_{k=1}^{n-1}2k\left(1+\displaystyle \sum_{k=1}^{n-1}2k \right)}{2}}[/tex]


Related Questions

P = {x:x is a multiple of 4, x ≤ 20} and Q = (x:x is a multiple of 5, x ≤ 20 } ​

Answers

Answer:

Hope it will help you.....

Step-by-step explanation:

my answer is

P=(4,8,12,16,20)

Q=(5,10,15,20)

The set for P and Q will be {4, 8, 12, 16, 20} and {5, 10, 15, 20}, respectively.

What are Sets?

Sets are groups of clearly specified components. The number of items in a finite set is denoted by a curly bracket.

P = {x:x is a multiple of 4, x ≤ 20}

Then the set for P is given as,

P = 5 {4, 8, 12, 16, 20}

Q = (x:x is a multiple of 5, x ≤ 20 } ​

Then the set for Q is given as,

Q = 4 {5, 10, 15, 20}

More about the sets link is given below.

https://brainly.com/question/8053622

#SPJ2

What is the equation of a line that passes through the point(3,-8)and has a slope of 0?

Answers

Answer: y=-8

Step-by-step explanation:

If the slope is zero that means that the line does not go up (or have any rise) and therefore must be a horizontal line. Since it passes through the point (3,-8), the equation has to be y=-8.

Riley's math teacher said that each question answered correctly on a test would be worth 3 points. Answer the questions below regarding the relationship between the number of questions correct and the score on the test.

Answers

The independent variable, x, represents the number of questions Riley answered correctly, and the dependent variable is the score, because the score depends on the number of questions Riley answered correctly. A function relating these variables is R(x) = [3x] So R(23) = [69], meaning 23 correct answers will give a score of 69.

The types of variables.

In an experiment, there are two (2) main types of variables and these include:

Dependent variable.Independent variable.

What is an independent variable?

An independent variable can be defined as the variable that is being manipulated by an experimenter or researcher. This ultimately implies that, it is typically considered to be the cause in an experiment and represented by x.

In this scenario, we can infer and logically deduce that the independent variable, x, represents the number of questions Riley answered correctly, and the dependent variable is the score, because the score depends on the number of questions Riley answered correctly. A function relating these variables is R(x) = [3x] So R(23) = [69], meaning 23 correct answers will give a score of 69.

Read more on independent variable here: https://brainly.com/question/82796

#SPJ1

Complete Question:

Riley's math teacher said that each question answered correctly on a test would be worth 3 points. Answer the questions below regarding the relationship between the number of questions correct and the score on the test.

The independent variable, x, represents the ______, and the dependent variable is the ______, because the ______ depends on the

A function relating these variables is R(x) = [] So R(23) = [], meaning 23

1) 2x = 12
solve the equation

Answers

Answer:

2x =12

x = 12/2

x= 6

I hope I helped you^_^

Answer:

x = 6

Step-by-step explanation:

2x = 12

x = 12/2

x = 6

Solve Y+3/5 = 1 1/10

Answers

Answer:

y = 1/2

Step-by-step explanation:

Y+3/5 = 1 1/10

Subtract 3/5 from both sides.

y = 1/2

Evaluate the function.
f(x) = -22 - 4.2
Find f(-1)

Answers

I hope this helps. Let me know if you can’t read something.

Find the perimeter. Simplify your answer.
SV
101-5
101-5
5v

Answers

2(10v - 5) + 5v + 5v =

= 20v - 10 + 5v + 5v =

= 20v + 5v + 5v - 10 = 30v - 10 ← the end

Please explain how to do this.

Answers

Answer:

hope this helps you

have a great

Maggie has a container in the shape of a right prism . The formula for its surface area is SA = Ph + 2B solve for h.
(9th grade Algebra 1)

Answers

Answer:

The choose first.

[tex] h = \frac{SA - 2B}{P} [/tex]

Step-by-step explanation:

[tex]SA = Ph + 2B \\ \\ Ph = SA - 2B \\ \\ h = \frac{SA - 2B}{P} [/tex]

I hope I helped you^_^

Grayson has $300 in a savings account. The interest rate is 10% per year and is not compounded. How much interest will he earn in 6 years?

Answers

Answer:

$180

Step-by-step explanation:

Interest for 1 year = 10% of $300

Interest for 6 years

= (10% × 6) of $300

= 60% of $300

= (60/100) × $300

= 60 × $3

= $180

So, he will earn $180 as interest in 6 years.

Write the equation of the line parallel to y=1/3x-6 that passes through the point (-12,3) in slope-intercept form.

Answers

Answer:

y=1/3x+7

Step-by-step explanation:

1. The given equation is y=1/3x-6 and it has to be parallel passing through (-12,3), so you just plug in the x and y values into the equation y=mx+b with 1/3 as the slope since its parallel (meaning the slope is the same for both eqts)

2. 3=1/3(-12)+b

3. 3=-4+b

4. 7=b

5. So, the equation is y=1/3x+7

please help 15 points

Answers

Answer:

2

Step-by-step explanation:

2x(5+4)=(_x5)+(_x4)

lets start with 2x(5+4)

(5+4)=9

so we are left with 2x9

2x9=18

so now we have 18=(_x5)+(_x4)

lets try 2 for the blank spaces

18=(2x5)+(2x4)

2x5=10

2x4=8

10+8=18

2x(5+4)=(2x5)+(2x4)

It worked! Hope this helps :)

Answer:

Both blanks equal 2.

Step-by-step explanation:

2*5=10

2*4=8

10+8=18

9*2=18

could someone please help me:) I am stick and I am not sure what to do ​

Answers

Answer:

Part 5.1.1:

[tex]\displaystyle \cos 2A = \frac{7}{8}[/tex]

Part 5.1.2:

[tex]\displaystyle \cos A = \frac{\sqrt{15}}{4}[/tex]

Step-by-step explanation:

We are given that:

[tex]\displaystyle \sin 2A = \frac{\sqrt{15}}{8}[/tex]

Part 5.1.1

Recall that:

[tex]\displaystyle \sin^2 \theta + \cos^2 \theta = 1[/tex]

Let θ = 2A. Hence:

[tex]\displaystyle \sin ^2 2A + \cos ^2 2A = 1[/tex]

Square the original equation:

[tex]\displaystyle \sin^2 2A = \frac{15}{64}[/tex]

Hence:

[tex]\displaystyle \left(\frac{15}{64}\right) + \cos ^2 2A = 1[/tex]

Subtract:

[tex]\displaystyle \cos ^2 2A = \frac{49}{64}[/tex]

Take the square root of both sides:

[tex]\displaystyle \cos 2A = \pm\sqrt{\frac{49}{64}}[/tex]

Since 0° ≤ 2A ≤ 90°, cos(2A) must be positive. Hence:

[tex]\displaystyle \cos 2A = \frac{7}{8}[/tex]

Part 5.1.2

Recall that:

[tex]\displaystyle \begin{aligned} \cos 2\theta &= \cos^2 \theta - \sin^2 \theta \\ &= 1- 2\sin^2\theta \\ &= 2\cos^2\theta - 1\end{aligned}[/tex]

We can use the third form. Substitute:

[tex]\displaystyle \left(\frac{7}{8}\right) = 2\cos^2 A - 1[/tex]

Solve for cosine:

[tex]\displaystyle \begin{aligned} \frac{15}{8} &= 2\cos^2 A\\ \\ \cos^2 A &= \frac{15}{16} \\ \\ \cos A& = \pm\sqrt{\frac{15}{16}} \\ \\ \Rightarrow \cos A &= \frac{\sqrt{15}}{4}\end{aligned}[/tex]

In conclusion:

[tex]\displaystyle \cos A = \frac{\sqrt{15}}{4}[/tex]

(Note that since 0° ≤ 2A ≤ 90°, 0° ≤ A ≤ 45°. Hence, cos(A) must be positive.)

Combine like terms to simplify the expression:

Answers

Answer:

[tex]\frac{5k -6}{10}[/tex]

Answer:

1/5k + 3/5

Step-by-step explanation:

2/5 * 2/2 = 4/10

4/10k + 1/10k = 5/10k = 1/2k

1/2k + 3/5

simplify expression x times 3/2 to the 6th power

Answers

Answer:

729x/64

Step-by-step explanation:

Expression: x *(3/2)⁶

Using order of operations (PEMDAS), we solve for the exponent first. So, 3^6 is 729 and 2^6 is 64. Multiple everything together to get 729x/64

Find the length indicated. label your answer with correct notation

Answers

Answer:

? is 8

Step-by-step explanation:

The total of the line is 14 and you have KL of your answer which is 6. All you have to do to get JK is subtract 6 from 14.

14 - 6 = 8

This one is very hard for me

Answers

15x + 57 ≥ 129

15x ≥ 129 - 57

15x ≥ 72

x ≥ 24/5

x 4.8

Answer:

[tex]x \geqslant 4.8[/tex]

Step-by-step explanation:

[tex]15x \: \: \: \: + \: 57 \geqslant 129 \\ or \: 15x \geqslant 129 - 57 \\ or \: x \geqslant 72 \div 15 \\ or \: x \geqslant 4.8 \\ [/tex]

Not sure how to find what x equals

Answers

Answer:

Its 48 °

Since its a triangle, the total angle will be 180°

So,

x° + 42°+90°=180°(90° as one angle in triangle is 90°)

x°=180°-90°-42°

x°=48°

Answer: 48º c:

Step-by-step explanation:  So basically the angle at the top, lets call it A, is 42º. And the angle at the bottom right, is a right angle, which is 90º. We know this by the square at the angle. If we call this angle B, then through the fact that triangles will always have a combined angle measuremnt of 180º, we can do a + b + x = 180, which will be 42 + 90 = 132, 180 - 132 = 48 :)

WILL GIVE BRAINLIEST
2x^\(5x + 5)
Find the value of x.​

Answers

25 is the value of x

if coffee cost $8.15 per pound what will it cost if it weights 2 pounds

Answers

Answer: $16.3

Step-by-step explanation: 8.15 * 2 (because there are 2 pound) = 16.3

Answer: it should be 16.30

Step-by-step explanation:

Can someone help me with this ASAP!

Answers

The following correct answer is c.

This I because when you translate using this, you first start with the point (0,0), translate that point down 3 (-3 means down three when outside parenthesis) and translate it to the right 4 units (this shown in a paraptosis, and when you move to the right you say -4, but if you move to the left you say +4)

the answer to the question is C

Write the equation of a line parallel to the following line that passes through (-5, 19).
y = 3x + 14

Answers

Lets find out slope

[tex]\\ \sf\longmapsto \dfrac{19-0}{0-(-5)}[/tex]

[tex]\\ \sf\longmapsto \dfrac{19}{5}[/tex]

Now

[tex]\\ \sf\longmapsto y=mx+b[/tex]

[tex]\\ \sf\longmapsto y=\dfrac{19}{5}x+19[/tex]

please help me simplify this answer explanation too thank!

Answers

[tex] - \frac{10}{3} - ( - \frac{7}{5} ) = - \frac{10}{3} + \frac{7}{5} = - \frac{50}{15} + \frac{21}{15} = \\ \\ = \frac{ - 50 + 21}{15} = \frac{ - (50 - 21)}{15} = \frac{ - 29}{15} = - 1 \frac{14}{15} [/tex]

Please just help with these questions!

Answers

5/7=        -----------

           0.714285

Convert the fraction to a decimal by dividing the numerator by the denominator.

6/10= 0.6

Convert the fraction to a decimal by dividing the numerator by the denominator.

16/20= 0.8

Convert the fraction to a decimal by dividing the numerator by the denominator.

9/10= 0.9

Convert the fraction to a decimal by dividing the numerator by the denominator.  

Write a inequality for the graph shown below. Use x for your variable

Answers

Answer:

[tex]x > 4[/tex]

Step-by-step explanation:

X is greater than 4

A map has a scale of 1 Inch to 64 miles. If the distance from Sacramento to Santa Cruz is 2.3 Inches long on the map, what is the actual distance in miles? (Round to the nearest mile)
A 28 miles
B 66 miles
C 147 miles
D 128 miles

Answers

Answer:

C

Step-by-step explanation:

I think this could be it since u have to multiple

64x2.3 Which would be 147.2 but then round up would get u 147

ers
17 Which number is greater, -7 or 6? Which number has the greater
absolute value, -7 or 6? Explain your thinking. Use comparison
symbols and absolute value symbols when you write your answer.
Solution

Answers

Answer:

17

Step-by-step explanation:

17-38 8 im like hey whats up hello

Intersecting lines are non-coplanar a.always b.sometimes c.never

Answers

The answer is
C. Never

angle abc and dcb make a linear pair. of abc=2x+9 and dbc=10x+15,find x

Answers

Answer:

x = 13

Step-by-step explanation:

A linear pair of angles sum to 180° , then

2x + 9 + 10x + 15 = 180 , that is

12x + 24 = 180 ( subtract 24 from both sides )

12x = 156 ( divide both sides by 12 )

x = 13

Please help.How do I find the length of x?

Answers

Since the area of B is 9 times the area of A, we know that the sides of B are √9= 3 times the length of the sides of A.

That's how.

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