I'm guessing you mean
[tex]S_{k+1} = S_k + a_{k+1}[/tex]
and that [tex]S_k[/tex] is the sum
[tex]S_k = 6 + 12 + 18 + \ldots + 6k = 6 (1 + 2 + 3 + \ldots + k) = 3 k (k+1)[/tex]
using the well-known formula
[tex]\displaystyle \sum_{i=1}^n i = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2[/tex]
Then by substitution,
[tex]S_{k+1} = 6 (1 + 2 + 3 + \cdots + k + (k+1)) = 3 (k+1) (k+2)[/tex]
and hence
[tex]a_{k+1} = S_{k+1} - S_k[/tex]
[tex]a_{k+1} = 3 (k+1) (k+2) - 3k (k+1)[/tex]
[tex]a_{k+1} = 3 (k+1) \bigg((k+2) - k\bigg)[/tex]
[tex]\implies \boxed{a_{k+1} = 6 (k + 1)}[/tex]
The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is four
times the measure of the first angle. The third angle is 12 more than the second. Let x, y, and z represent the measures of
the first, second, and third angles, respectively. Find the measures of the three angles.
Answer:
x = 36°
y = 66°
z = 78°
Step-by-step explanation:
General outline:Setup equations for each sentenceSolve the resulting systemSetup equations:
Allowing the first angle to be x, the second to be y, and the third to be z, the three sentences describing the relationships between angle measures are given below:
Eq1: x + y + z = 180°Eq2: y + z = 4xEq3: z = y + 12°Solve the systemSubstituting equation 3, into equations 1 and 2
Eq1: x + y + z = 180°
x + y + (y+12°) = 180°
x + 2y + 12° = 180°
x + 2y = 168°
x = 168° - 2y
Eq2: y + z = 4x
substituting the result from equation 1, and Equation 3...
y + (y+12°) = 4*(168° - 2y)
2y + 12° = 672° - 8y
10y = 660°
y = 66°
Substituting back into Equation 3...
Eq3: z = y + 12°
z = (66°) + 12°
z = 78°
Substituting both y, and z into Equation 1...
x + y + z = 180°
x + (66°) + (78°) = 180°
x + 144° = 180°
x = 36°
So, the three angle measures are:
x = 36°
y = 66°
z = 78°
Verification"The sum of the measures of the angles of a triangle is 180."
x + y + z = 180°
36° + 66° + 78° = 180°
True
"The sum of the measures of the second and third angles is four times the measure of the first angle"
y + z = 4x
66° + 78° = 4*(36°)
144° = 144°
True
"The third angle is 12 more than the second"
z = y + 12°
78° = 66° + 12°
True
The probability all of the students in this class will be present and on time is 3/10. The probability all of the students are present is 2/5. What is the probability that everyone is on time given that everyone is present? Give your answer as a simplified fraction.
Answer:
3/4
Step-by-step explanation:
A: students present
B: students on time
P(all students present and on time) = P(A and B) = 3/10
P(all students present) = P(A) = 2/5
P(A and B) = P(A).P(B|A) where P(B|A) is the probability of everyone being on time given that everyone is present
So P(B|A) = P(A and B) /P(A) = 3/10 ÷ 2/5 = 3/10 * 5/2 = 15/20 which can be reduced to 3/4 by dividing numerator and denominator by 5
How many significant figures does 119.3 have
Please help me....trying to get my HS diploma, i did not graduate :(
A ball is thrown in air and it's height, h(t) in feet, at any time, t in seconds, is represented by the equation h(t)=−t2+7t. When is the ball higher than 10 feet off the ground?
Answer:
A.
Step-by-step explanation:
so, the equation is
h(t) = -t² + 7t
so, we need to find the solutions for t (the time when the ball is exactly 10 ft in the air). there had to be 2 solutions, as the ball first goes up passing the 10 ft height, and then comes back down again, passing the 10 ft mark a second time. and between these 2 times the ball is higher (but not equal, so, we can only use < or > as inequality signs) than 10 ft.
10 = -t² + 7t
-t² + 7t - 10 = 0
the generation solution to a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = t
a = -1
b = 7
c = -10
t = (-7 ± sqrt(7² - 4×-1×-10))/(2×-1) =
= (-7 ± sqrt(49 - 40))/-2 = (-7 ± sqrt(9))/-2
t1 = (-7 + 3)/-2 = -4/-2 = 2 seconds
t2 = (-7 - 3)/-2 = -10/-2 = 5 seconds
so, between 2 and 5 seconds airtime the ball is higher than 10 ft.
and remember : HIGHER THAN.
so, we cannot use any equality (like <= or >=).
t must be higher than 2 and lower than 5 :
2 < t < 5
graph y(t) = 13.5t - 4.9t^2
Answer:
here
Step-by-step explanation:
Which
graph shows a dilation?
Suppose 4y – 1 = 15 and y = 5x – 3.
Which of the following equations is equivalent to 4y – 1 = 15, but written only in terms of x?
The equivalent equation of 4y – 1 = 15 is 20x - 13 = 15
How to determine the equivalent equation?The equations are given as:
4y – 1 = 15 and y = 5x – 3.
Substitute y = 5x – 3. in 4y – 1 = 15
4(5x – 3) – 1 = 15
Expand
20x- 12 - 1 = 15
Evaluate the like terms
20x - 13 = 15
Hence, the equivalent equation of 4y – 1 = 15 is 20x - 13 = 15
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The sum of two numbers is 32. One number is 3 times as large as the other. What are the numbers?
Answer:
Let the greater number be x and the smaller number be y
A.T.Q,
x+y=32 ...(1)
x=3y ...(2)
putting the value of x from (2) in (1)
(3y)+y=32
4y=32
y= 32/4=8
x=3y
x=3(8)=24
x=24 , y=8
Dont Really need explanation Simply answer
Considering the given graph, the profit during the period was of -2 dollars.
How to add negative numbers along with positive?We add all the negative numbers, all the positive numbers, then keep the signal of the higher absolute value, and subtract the values.
From the graph, we have that:
The sum of the negative numbers is of 8.The sum of the positive numbers is of 2 + 4 = 6.8 > 6, hence the profit is negative given by:
-(8 - 6) = -2 dollars.
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A can do a piece of work in 40 days. He works at it for 5 days and then B finishes it in 10 days.
In what time can A and B together finish it ?
Answer:Given
a can do a piece of work = 40 days
so, A's 1 day work =1/4
so, A's 8 day's work =8/40=1/5
remaining work = 1- 1/5= 4/5
Now, B can do 4/5 piece of work in =16 days
So, B’s 1 day's work = (4/5) /16= 4/ 5x6=1/20
Therefore, (A+B)’s 1 day's work = 1/40+1/20 =
1+2/40=3/40
∴ Both (A+B) can do a piece of work in =
1/days (3/40) = 40/3 days
= 13 1/3 days
The following stem-and-leaf plot represents the test scores for
26 students in a class on their most recent test. Use the data provided to find the quartiles.
Stem
Leaves
6 0 1 2 2 4 7 8
7 0 1 2 5 8
8 3 6 6 7 9 9
9 0 1 2 2 2 4 5 8
Key:
6|0=60
To find the second quartile, we must first divide the data in half. There are an even number of data values, so the 50th percentile, or median, is the arithmetic mean of the two values in the middle of the set. Thus, the median is the mean of the 13th data value,
83 and the 14th data value, 86
83+86/2=84.5
Therefore, the second quartile is 84.5
Step 2 of 3 : Find the first quartile.
Considering the given stem-and-leaf plot, the quartiles are given as follows:
The first quartile is of 67.5.The second quartile, which is the median, is of 84.5.The third quartile is of 91.5.What are the median and the quartiles of a data-set?The median of the data-set separates the bottom half from the upper half, that is, it is the 50th percentile.The first quartile is the median of the first half of the data-set.The third quartile is the median of the second half of the data-set.There is an even number of elements(26), hence the median is the mean of the 13th and 14th elements, which are 83 and 86, hence:
Me = (83 + 86)/2 = 84.5.
The first half has 12 elements, hence the first quartile is the mean of the 6th and 7th elements, which are 67 and 68, hence:
Q1 = (67 + 68)/2 = 67.5.
The third half also has 12 elements, starting at the second 86, hence the third quartile is the mean of the 6th and 7th elements of this half, hence:
Q3 = (91 + 92)/2 = 91.5.
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3,5,12,55,648? vwhat is the next number
Hello,
Answer:
The next number is 35 585
Step-by-step explanation:
3 × 5 - 3 = 12
5 × 12 - 5 = 55
12 × 55 -12 = 648
55 × 648 - 12 = 35 585
The next number for the given series is 35,585.
What is number series?A number series is a number-based sequence. In the number series questions, we must identify the rules that lead to the formation of a number series. These series vary in nature, as do the questions that may arise from them.
Given that the series is 3,5,12,55,648. The next number will be calculated by decoding the series as below,
3 × 5 - 3 = 12
5 × 12 - 5 = 55
12 × 55 -12 = 648
55 × 648 - 12 = 35,585
Therefore, the value of the next number for the series is 35,585.
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Choose the number sentence that applies the commutative property to the example.
3 × 4 = 12
6 + 6 = 12
12 × 1 = 12
6 × 2 = 12
4 × 3 = 12
! PLS HELP ! There are a total of 30 white and black marbles in a box. There are 14 more white marbles than black marbles. How many black marbles are in the box?
Answer:
There are 8 black marbles.
Step-by-step explanation:
First, set up your equation. You have black (b) and white (w) marbles, total of 30, so those will be your variables to set in your equation.
w + b = 30
You mentioned that there are 14 more white marbles then black. Therefore, white marbles will have b many marbles + 14, creating the equation:
w= b + 14
If w = b + 14 and w can be easily substituted in the first equation, substitute w for b + 14. This will result in having the equation of:
(b +14) + b= 30
Simplify this equation, combine like terms to break down the equation, making it easier for you to solve for b.
-Subtract 14 from both sides.
-Add b + b = 2b
Then, you will have your resulting equation in which you divided 2 from both sides to get b by itself.
2b = 16
b=8
There are 8 black marbles.
A box contains 162 marbles which are either red or blue. There are 42 more blue marbles than red marbles. How many blue marbles are in the box? Blue marbles=
The number of the blue marbles are in the box will be 102.
What is the solution of the equation?The solution of the equation means the value of the unknown or variable.
A box contains 162 marbles which are either red (r) or blue (b).
r + b = 162 ...1
There are 42 more blue marbles than red marbles.
b = r + 42 ...2
Then from equation 1 and 2, we have
r + r + 42 = 162
2r = 120
r = 60
Then the number of blue marble will be
b = 60 + 42
b = 102
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The boxed definition of absolute value states that |a|=-a if a is a negative number. Explain why |a| is always nonnegative, even though |a|=-a for negative values of a.
Final answer:
Since absolute values determine the distance between the number and the value whether the value is positive or negative. As distance is always positive.
Thus, |a| is always nonnegative, even though |a|=-a for negative values of a.
Step-by-step explanation:
Step 1
It is said that |a| is always nonnegative even though even though |a|=-a for negative values of a.
Step 2
This is because by the definition of an absolute value, any real number inside an absolute value symbol || will always be positive.
Over the past five years, an analyst records the annual returns (in percent) for a mutual fund as: 13, −2, 3, 18, and −14. Construct the 95% confidence interval for μ.
The required confidence interval for μ is (-12.07, 19.27 )
Annual returns (in percent) for a mutual fund as: 13, −2, 3, 18, and −14
95% Confidence interval
The statistic is the study of mathematics which deal with relations between comprehensive data.
Mean(x) = 13-2+3+18-14/5
x = 3.6
Standard deviation = [tex]\sqrt{\frac{\sum(x_i- x)^2}{n-1} }[/tex]
S = 12.6214
Point estimate of (μ) = 3.6
Confidence interval = mean (+,-) Tc x S/√5
= 3.6 (+,-) 2.7764 x 12.62/√5
=3.6 (+,-) 15.671
= (-12.07, 19.27)
Thus, the required confidence interval for μ is (-12.07, 19.27 ).
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On a coordinate plane, a line goes through (negative 4, 0) and (0, 2). The graph of the equation y = 12 x + 2 is displayed. Which equations would intersect the orange line at the y-intercept? Check all that apply. y = 7x + 2 y = 3x + 5 y = 2 minus StartFraction 7 Over 3 EndFraction x y – 2 = 4x 3y = –9x – 6
The equation of line which would intersect be
y = 7x + 2 y = 2- 7/3x y-2 = 4xWhat is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Given:
y = 12 x + 2
and, a line passes through (-4, 0) and (0, 2)
Equation of line
y = -1/2 (x+ 4)
y= -x/2 - 2
The equations which would intersect the line will be
y = 7x + 2 y = 2- 7/3xy-2 = 4xLearn more about equation here:
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x, 3x and 5x ,find the sum
x + 3x + 5x ( the variables are same you can add them immediately )
9x ( the answer is 9x for adding terms with the same variable just add the numbers and write the variable as itself)
HOPE IT HELPS
Answer: 9x
Step-by-step explanation:
As the variables are of same power (1) we can add it
x+3x+5x=9x
A baker purchased 13 lb of wheat flour and 14 lb of rye flour for a total cost of $14.85. A second purchase, at the same prices, included 15 lb of wheat flour and 12 lb of rye flour. The cost of the second purchase was $14.85. Find the cost per pound of the wheat flour and of the rye flour.
An equation is formed when two equal expressions. The cost per pound of wheat flour is $0.55 and the cost of per pound of rye flour is $0.55.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
Let the cost of a pound of wheat flour be x and the cost of a pound of rye flour be y.
A baker purchased 13 lb of wheat flour and 14 lb of rye flour for a total cost of $14.85.
13x + 14y = $14.85
x = (14.85-14y)/13
A second purchase, at the same prices, included 15 lb of wheat flour and 12 lb of rye flour.
15x + 12y = $14.85
Substitute the value of x from the above equation and solve for y,
[tex]\dfrac{15(14.85-14y)}{13} + 12y = 14.85\\\\222.75-210y+156y = 193.05\\\\y = 0.55[/tex]
Substitute the value of y in any one of the equations formed,
13x + 14(0.55) = 14.85
13x = 7.15
x = 0.55
Hence, the cost per pound of wheat flour is $0.55 and the cost of per pound of rye flour is $0.55.
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solving 3 x plus 8 equals 14
Choose the best selection for the
quadrilateral with vertices at the
following points:
(1,0), (1,4), (6,0), (6,4)
Hint: Start by graphing the points.
Distance Formula: d= √(x2-x1)² + (y2 − ₁)²
A. Rectangle
C. Rhombus
B. Square
D. Trapezoid
Answer: a rectangle
Step-by-step explanation:
If you graph the points out you will see that it is four points high and six points wide
Answer:
It is A.Rectangle. from the given points.
Step-by-step explanation:
the graph is given in the image.
I hope it to be helpful ^=^
(02.02 LC)
If h(x) = −2x − 10, find h(−4).
Group of answer choices
−2
−18
−3
−16
The value of the function h(-4) is (a) -2
How to determine the h(-4)?The function is given as
h(x) = -2x - 10
When x = -4, we have:
h(-4) = -2 * -4 - 10
Evaluate the product
h(-4) = 8 - 10
Evaluate the difference
h(-4) = -2
Hence, the value of the function h(-4) is (a) -2
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The height of a triangle is 4 m less than its base. The area of the triangle is 48 m². What is the length of the base? Enter your answer in the box.
The length of the base of the triangle is 12m
Area of a triangleThe formula for calculating the area of a triangle is expressed as:
A = 0.5 * base * height
If the height of a triangle is 4 m less than its base, then;
h = b - 4
Substitute
A = 0.5(b)(b-4)
48 = 0.5b²-2b
b²-4b - 96 = 0
Factorise
b² - 12b + 8b - 96 = 0
b(b-12)+8()b-12) = 0
b - 12 = 0
b = 12
Hence the length of the base of the triangle is 12m
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Scientists want to test a new pair of running shoes. A speed test is performed with two separate groups of participants. The treatment group will wear the new pair of running shoes, while the control group will not. It is believed that age and height may affect speed. Which of the following would be most effective in controlling the confounding variables, such as age and height, in this study?
The most effective in controlling the confounding variables is matched pairs design.
What is a matched pair design experiment?A matched pairs design is when participants are matched based on shared characteristics that is of importance to the experiment. One member of the pair is put in the control group while the other is put in the experimental group.
Here are the options:
A completely randomized design experiment
A longitudinal observational study
A retrospective observational study
A matched-pair design experiment
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Is the relationship between red sox away games and average daily revenues significant at the 95% confidence level? choose the correct answer with the corresponding correct reasoning.
Answer:
Yes, because the p-value of the independent variable is less than 0.05
Step-by-step explanation:
The relationship between Red Sox away games and average daily revenues significant at the 95% confidence level because the p-value ( X) < 0.05 , Option C is the correct answer.
What is Probability ?Probability is the study of chances of an event to happen.
It has a range form 0 to 1 , where 0 indicates uncertainty while 1 indicates certainty.
The data is given and on the basis of it
It can be seen that for 95 % confidence Level
P value =0.0005
So , the relationship between Red Sox away games and average daily revenues significant at the 95% confidence level because the p-value of the independent variable is less than 0.05.
Therefore Option C is the correct answer.
The complete question is
Is the relationship between Red Sox away games and average daily revenues significant at the 95% confidence level? Choose the correct answer with the corresponding correct reasoning.
No, because R Square is less than 50%
Yes, because the slope is positive.
Yes, because the p-value of the intercept is less than 0.05
Yes, because the p-value of the intercept is less than 0.95
Yes, because the p-value of the independent variable is less than 0.05
Yes, because the p-value of the independent variable is less than 0.95
The data table is attached with the answer.
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Two cars leave town at the same time one travels West at 56 miles per hour and the other travel south at 42 miles per hour after 1 hour how far apart are the cars
What is a possible value for the missing term of the geometric sequence? 25,_,874,…
Answer:
The middle term is 5√874======================
Given sequence25, x, 874, ...To findThe missing term xSolutionSince the given is the GP, find the common ratio in terms of given values:
r = x/25r = 874/xEquate the two values and solve for x to find the missing term:
x/ 25 = 874/xx² = 874*25x² = 21850x = √21850x = 5√874Kate is the captain of a large cruise ship. As she
drives the ship, she sits 50 meters above the sea.
Paul is the captain of a boat. He sits 15 meters
above the sea. If the radius of the earth is
6,371,000 meters, how much further can Kate see
across the horizon than Paul?
here are the first few questions the rest are in the picture.
1.Show me that you understand the problem. restate it in your own words and identify the important information.
2. what is your prior knowledge and what tools do you have to solve this problem?
a. Draw two diagrams, one for Kate and one for Paul. Include important numbers. Insert your diagrams into your answer.
The amount further by which Kate can see across the horizon than Paul is; 3.33 times farther
How to use volume of a cone in applications?Formula for volume of a cone is;
V = ¹/₃πr²h
For Kate:
Height (h) = 50 m
Radius (r) = 6,371,000 m
Thus;
V = ¹/₃ * π * (6,371,000)² * 50
V = 2.12526863 × 10¹⁵ m³
To determine how much further can kate see across the horizon than Paul, we have:
⇒ (2.12526863 × 10¹⁵ m³)/6.3758059 × 10¹⁴ m³
= 3.33 times further.
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Select the correct answer.
What is the solution to this equation?
(z
(x + 6) - 5 = -2
OA.
OB. 7
OC. 21
OD. -139
Answer:
-3
Step-by-step explanation:
Equation: (x + 6) - 5 = -2
we want to isolate x
(x + 6) - 5 = -2
+ 5 + 5 (add 5 to start getting x on its own)
x + 6 = 3
- 6 - 6 (subtract 6 to isolate x)
x = -3
so, this answer was not provided, but is the correct answer.
The solution to this equation is -3,
so, the correct answer is: -3.
Here, we have,
given that,
Equation is:
(x + 6) - 5 = -2
we want to isolate x
(add 5 to start getting x on its own)
(x + 6) - 5+ 5 = -2 + 5
(subtract 6 to isolate x)
x + 6 - 6 = 3 - 6
so, we get,
x = -3
so, the solution to this equation is -3,
so, the correct answer is: -3.
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