Answers
Since d(1) = 1, we can assert that the induction proves that the value of n is true for n = 1, and there is only one dot. Also, the mathematical induction is true for all natural numbers (n).
What is mathematical induction?A mathematical approach known as mathematical induction is used to demonstrate that a claim or statement is true for each and every natural number.
You must demonstrate the following proposition using an inductive step of proof such that d(k) and d(k+1) are both true if d(k) is true.
To Prove:
\(\mathbf{d(n) = \dfrac{n(n+1)}{2}}\)
For n = 1
\(\mathbf{d(1) = \dfrac{1(1+1)}{2}}\)
\(\mathbf{d(1) = \dfrac{1 \times 2}{2}}\)
\(\mathbf{d(1) = \dfrac{2}{2}}\)
d(1) = 1 this is true for n = 1, there is only one dot.
b.
Now, let us assume that the result is true for n = k + 1 such that:
\(\mathbf{d(k+1) = \dfrac{(k+1) ((k+1) + 1) }{2}}\)
\(\mathbf{d(k+1) = \dfrac{(k+1) (k+2) }{2} ---- (1)}\)
Now, Let n = k, then:
\(\mathbf{d(k) = \dfrac{k(k + 1)}{2} ----(2)}\)
Recall that, the total number of dots d(n) increases by 'n' each time.
i.e. d(n+1) = d(n) + n
Therefore;
d(k) = d(k+1) + k
To prove:
\(\mathbf{\dfrac{k(k + 1)}{2} + (k + 1) = \dfrac{(k+1) (k+2) }{2}}\)
\(\mathbf{ \dfrac{k(k+1)+ (2k+2) }{2}= \dfrac{(k+1) (k+2) }{2}}\)
\(\mathbf{ \dfrac{(k+1) (k+2) }{2}= \dfrac{(k+1) (k+2) }{2}}\)
L.H.S = R.H.S
Therefore, we can conclude that the mathematical induction is true for all natural numbers (n).
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Related Questions
what is the solution for 3х + y = -5
Answers
\(\tt Step-by-step~explanation:\)
\(\tt Solve~for~y:\)
To solve for y, we have to move all of the terms that do not equal to y on one side, and let y be on the other side alone. Do this by subtracting 3x on both sides to cancel it out from the left side and bring it to the right.
\(\tt 3x-3x+y=-5-3x\\y=-5-3x\\y=-3x-5\)
\(\tt Solve~for~x:\)
To solve for x, we will do the same thing we did to solve for y: move all the terms that do not equal to x to one side, and leave x on another side of the equation. We can do this by first subtracting y from both sides.
\(\tt3x+y-y=-5-y\\3x=-y-5\)
Then, we divide all terms by 3 to isolate the x.
\(\tt \frac{3x}{3}=x\\\\ \frac{-y}{3}=\frac{-y}{3} \\\\\frac{-5}{3}=\frac{-5}{3}\)
\(\tt x=-\frac{y}{3} -\frac{5}{3}\)
\(\lare\boxed{\tt Our~final~answer:~y=-3x=5,~x=-\frac{y}{3}-\frac{5}{3} }\)
PLEASEE HURRYY Which expression is equivalent to the complex number. 10+3i
A) 2i(4-5i)+(1-7i)
B) (4+7i)-2i(2+3i)
C) (-3+5i)-3i(4+5i)
D) 3i(4+7i)+(11+2i)
Answers
Answer:
(4+7i)-2i(2+3i) = 10+3i
Step-by-step explanation:
We need to find the expression that is equivalent to the complex number 10+3i.
Option 1. 2i(4-5i)+(1-7i)
=8i-10i²+1-7i
∵ i² = -1
=8i-10(-1)+1-7i
=8i+10+1-7i
=11+i (incorrect)
Option 2. (4+7i)-2i(2+3i)
=4+7i-4i-6i²
=4+7i-4i-6(-1)
=4+7i-4i+6
=10+3i (Correct)
Option 3. (-3+5i)-3i(4+5i)
= (-3+5i)-12i-15i²
= -3+5i-12i-15(-1)
= -3+5i-12i+15
=12-7i (incorrect)
Option 4. 3i(4+7i)+(11+2i)
= 12i+21i²+11+2i
=12i+21(-1)+11+2i
= 12i-21+11+2i
=14i-10 (incorrect)
Hence, the correct option is (B).
Answer:
the answer is for me A they might be lettered different for each
Step-by-step explanation:
At a town meeting, the ratio of dark-haired people to blond-haired people to red-haired people is 42 : 37 : 3. If there are 1,312 people at the meeting, how many have each color hair?
Answers
Answer:
672 had dark hair, 592 had blond hair, and 48 had red hair
Step-by-step explanation:
To solve this problem, we need to first find the total number of people for each hair color based on the given ratio.
Let's start by finding the common factor that we can use to scale the ratio up to the total number of people, which is 1,312:
42 + 37 + 3 = 82
We can then divide 1,312 by 82 to get the scaling factor:
1,312 ÷ 82 = 16
This means that for every 16 people, there are 42 with dark hair, 37 with blond hair, and 3 with red hair.
To find the actual number of people with each hair color in the town meeting, we can multiply the scaling factor by the number of people for each hair color in the ratio:
Dark-haired people: 42 × 16 = 672
Blond-haired people: 37 × 16 = 592
Red-haired people: 3 × 16 = 48
Therefore, there are 672 people with dark hair, 592 people with blond hair, and 48 people with red hair at the town meeting.
2
Cruz is the editor of a magazine.
The magazine has 40 pages.
The magazine must have
.
.
20% of the pages for adverts
1/4 of the pages for photos
the remaining pages for articles.
.
Work out the number of pages for articles in the magazine.
(4 Points
Answers
Answer:
40 pages
20% is 8 pages
1/4 is 10
22 pages for articles
I kind of understand but what I'm having trouble with is when u simply it 6/5 ×5/6=6/5×40 how are they getting 48 i get 33.3
Answers
37 = -3 + 5(x + 6)
Distributing:
37 = -3 + 5(x) + 5(6)
37 = -3 + 5x + 30
Commuting:
37 = -3 + 30 + 5x
37 = 27 + 5x
27 is adding on the right, then it will subtract on the left.
37 - 27 = 5x
10 = 5x
5 is multiplying on the right, then it will divide on the left
10/5 = x
2 = x
health programs routinely study the number of days that patients stay in hospitals. in one study, a random sample of 12 men had a mean of 7.95 days and a standard deviation of 6.2 days, and a random sample of 19 women had a mean of 7.1 days and a standard deviation of 5.0 days. the sample data will be used to construct a 95 percent confidence interval to estimate the difference between men and women in the mean number of days for the length of stay at a hospital. have the conditions been met for inference with a confidence interval?
Answers
The conditions for inference with a confidence interval have been met, and we can proceed to construct a confidence interval to estimate the difference between men and women in the mean number of days.
What is confidence interval ?
A confidence interval is a range of values that is likely to contain the true value of a population parameter with a certain level of confidence.
To determine if the conditions for inference with a confidence interval have been met, we need to check the following assumptions:
Random Sampling: The sample of men and women should be random and representative of the population.Normality: The distribution of the difference between the means should be approximately normal.Independence: The two samples should be independent of each other.Since the sample sizes are greater than 30 for both men and women, we can assume normality based on the Central Limit Theorem.
Also, since the samples are selected independently and randomly, the assumption of independence is met.
Therefore, the conditions for inference with a confidence interval have been met, and we can proceed to construct a confidence interval to estimate the difference between men and women in the mean number of days for the length of stay at a hospital.
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what is the name for a data value that is far above or below the rest?
Answers
The name for a data value that is far above or below the rest is called an outlier.
An outlier is an observation that deviates significantly from other observations in a dataset. It is an extreme value that lies outside the typical range of values and may have a disproportionate impact on statistical analyses and calculations. Outliers can occur due to various reasons, including measurement errors, data entry mistakes, or genuine rare events. Identifying and handling outliers appropriately is important in data analysis to ensure accurate and reliable results.
When dealing with outliers, it is important to assess whether they are the result of errors or genuine extreme values. Statistical techniques, such as box plots, scatter plots, or z-scores, can be used to detect outliers. Once identified, the appropriate action depends on the nature and cause of the outliers. In some cases, outliers may need to be corrected or removed from the dataset, while in other cases, they may provide valuable insights or require further investigation.
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I need the answer to all of them Rolly quick Bc my phone is dying
Answers
Answer:
1: $5
2: 7
3: 18
4: 17
5: 2
6: 1,029
Step-by-step explanation:
If in 2003 a girl was 6 years of age then what year is it , now that she is 22?
Answers
Tina wrote the equations 3 x minus y = 9 and 4 x + y = 5. What can Tina conclude about the solution to this system of equations?
Answers
Answer:
(2, –3) is a solution to the system of linear equations.
Step-by-step explanation:
Given: Equations:
3x - y = 9 --------(1),
4x + y = 5 --------(2),
Add Equation (1) + Equation (2),
3x + 4x = 9 + 5
7x = 14 ( Combine like terms )
x = 2 ( Divide both sides by 7 ),
From equation 1:
3(2) - y = 9
6 - y = 9
-y = 9 - 6 ( Subtraction 6 on both sides )
-y = 3
y = - 3 ( Multiplying -1 on both sides )
A sampling distribution is normal only if the population is normal. Question content area bottom Part 1 Choose the correct answer below. A. The statement is true. B. The statement is false. A sampling distribution is never normal. C. The statement is false. A sampling distribution is normal if either n30 or the population is normal. D. The statement is false. A sampling distribution is normal only if n30.
Answers
The statement "A sampling distribution is normal if either n≥30 or the population is normal" is true.
Hence option C is correct.
It is ideal for the population to be normal, a larger sample size (n≥30) can also result in a normal sampling distribution due to the Central Limit Theorem.
Therefore, a normal sampling distribution is not never possible as stated in option B, and it's not solely dependent on the population being normal as stated in option A.
Option D, stating that a normal sampling distribution is only possible if n≥30 is also not entirely true, as a normal population can still result in a normal sampling distribution for smaller sample sizes.
Hence the correct statement is,
"A sampling distribution is normal if either n≥30 or the population is normal".
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A test of the hypotheses H0: p = .25 versus Ha: p > .25 provides a p-value of 0.11.
Answers
Based on the provided information, if a test of the hypotheses H0: p = .25 versus Ha: p > .25 provides a p-value of 0.11, we can conclude that there is not enough evidence to reject the null hypothesis at a significance level of .05.
since the p-value is greater than the level of significance. However, we cannot completely rule out the possibility of a true difference existing between the sample proportion and the hypothesized proportion, as the p-value is not very small.
Based on the provided information, you conducted a hypothesis test with the null hypothesis (H0) stating that the proportion (p) is equal to 0.25, and the alternative hypothesis (Ha) stating that the proportion (p) is greater than 0.25. The test resulted in a p-value of 0.11.
To determine whether to accept or reject the null hypothesis, you'll need to compare the p-value to a predetermined significance level (alpha). If the p-value is less than or equal to alpha, you would reject the null hypothesis in favor of the alternative hypothesis. If the p-value is greater than alpha, you would fail to reject the null hypothesis.
Without a specified significance level, it's not possible to make a definitive conclusion. However, if using a common alpha level of 0.05, you would fail to reject the null hypothesis since the p-value (0.11) is greater than alpha (0.05).
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There are 100 marbles in the bag. John picks 10 marbles
out of the bag, and 6 of them are red.
What would be an estimate for the number of red marbles
in the bag?
Answers
Answer:
60 red marbles
Step-by-step explanation:
If John picks 10 marbles and 6 of them are red, there is a 60% chance of pulling out a red, which also means 60% of the total number of marbles are red.
(To be fair, 10 marbles is not a lot to base an estimate on, but I digress.)
Therefore, if there is a total of 100 marbles in the bag, 60% of 100 is 60. Our estimate will be 60 red marbles.
Answer:
60 red marbles
Step-by-step explanation:
10 marbles are taken out of the bag.
6 marbles taken out are red.
So the other 4 marbles taken out are blue.
200/4 = 50
50 is close to 60.
Match each expression with an equivalent radical form. One of the radical form
options will not have a match.
3 7/2
7 1/2
3 2/7
7 2/3
1. √7
2. √37
3. √3²
4. 3/72
Answers
Answer:
3^(7/2) = 2.; 7^(1/2) = 1,; 3^(2/7) = 3.; 7^(2/3) = 4.
Step-by-step explanation:
To convert the expressions to their radical form, we remember this formula:
\(b^\frac{exponent}{index}=\sqrt[index]{b^e^x^p^o^n^e^n^t}\)
Thus, for 3^(7/2), 7 is the exponent and 2 is the index, which gives us
\(\sqrt[]{3^7}\).
For 7^(1/2), 1 is the exponent and 2 is the index, which gives us
\(\sqrt{7}\).
For 3^(2/7), 2 is the exponent and 7 is the index, which gives us
\(\sqrt[7]{3^2}\).
For 7^(2/3), 2 is the exponent and 3 is the index, which gives us
\(\sqrt[3]{7^2}\)
Kira had 15 pencils. She gave 13 of the pencils to her friend. How many pencils did she give her friend?
Answers
Answer:
13
Step-by-step explanation:
it says she gave 13 to her friend
If IR is 120°, how many sides does the figure have?
Answers
The number of sides of the regular figure is 6.
How many sides does the polygon have?Each interior angle of a regular polygon is expressed as:
Interior angle = (( n - 2) × 180°) ÷ n
Where n is the number of sides of the polygon.
Given that:
One interior angle = 120 degrees
Number of sides n = ?
Plug the given value into the above formula and solve for number of sides n:
Interior angle = (( n - 2) × 180°) ÷ n
120° = (( n - 2) × 180°) ÷ n
Cross multiply:
120n° = ( n - 2 ) × 180°
120n° = 180°n - 360°
180°n - 120n° = 360°
60n° = 360°
n = 360° / 60°
n = 6
Therefore, the number of is 6.
This question is incomplete, the complete question is:
Find the Number of Sides of a Regular Polygon if One Interior Angle is 120 degrees
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Peter has half of his investments in stock paying a 6% dividend and the other half in a stock paying 14 % interest. If his total annualinterest is $500, how much does he have invested?
Answers
Assume that Peter will invest $x
Then he will invest 1/2 x with a rate of 6% and 1/2 x with a rate of 14%
Since the rule of the interest is
\(I=\text{PRT}\)P is the amount of investment
R is the rate in decimal
T is the time
For the first account
\(\begin{gathered} P=\frac{1}{2}x \\ R=\frac{6}{100}=0.06 \\ T=1 \end{gathered}\)For the second account
\(\begin{gathered} P=\frac{1}{2}x \\ R=\frac{14}{100}=0.14 \\ T=1 \end{gathered}\)Then the interest of each account is
\(\begin{gathered} I_1=(\frac{1}{2}x)(0.06)(1) \\ I_1=0.03x \end{gathered}\)\(\begin{gathered} I_2=(\frac{1}{2}x)(0.14)(1) \\ I_2=0.07x \end{gathered}\)Since the total interest is $500, then add I1 and I2, equate the sum by 500
\(\begin{gathered} I_1+I_2=500 \\ 0.03x+0.07x=500 \\ 0.10x=500 \end{gathered}\)Divide each side by 0.10
\(\begin{gathered} \frac{0.10x}{0.10}=\frac{500}{0.10} \\ x=5000 \end{gathered}\)Then he invested in each account 5000/2 = $2500
He invested a total of $5000
given a triangle which angle A=72 degrees, B=27. what would be the value for angle C
Answers
Answer:
81°
Step-by-step explanation:
add both and subtract answer from 180
72+27=99
180-99=81
all 3 angles in a triangle always add up to 180°
hope this helps <3
While reading about a research study, which of the following would tell you that an association claim is being
made?
a. The presence of a scatterplot or bar graph
b. The measurement of two variables
c. The use of a correlation coefficient
d. The interrogation of internal validity
Answers
The answer to this question is c.The use of a correlation coefficient.
The use of a correlation coefficient would tell you that an association claim is being made in a research study. A correlation coefficient is a statistical measure that shows the strength and direction of a relationship between two variables. It ranges from -1 to 1, with 0 indicating no relationship and -1 or 1 indicating a perfect negative or positive relationship, respectively.
A scatterplot or bar graph may be used to visually display the relationship between two variables, but they do not necessarily indicate that an association claim is being made. The measurement of two variables is necessary for any type of research study, but it does not inherently imply that an association claim is being made.
Interrogation of internal validity is a process used to ensure that the study's results accurately reflect the relationship between the variables being studied. It is important for any research study but does not specifically indicate that an association claim is being made.
Therefore,the answer to this question is c.The use of a correlation coefficient.
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Help plz ...............
Answers
Answer:
<RQP
Step-by-step explanation:
The vertex must either be the name of the angle or in the middle of the letters
<PQR or <RQP
We can also just call it <Q since there is only one angle
A marketing firm obtained random samples of 20 people in five regions of the country to investigate the level of interest in a new product. People in the sample were asked to rate their level of interest on a scale from 1 to 10, with 1 being the least amount of interest and 10 being the greatest. The histograms show the results for each region. The graph for which region displays data for level of interest with the least standard deviation?.
Answers
Without the histograms, there's no specific region as an answer. However, you can follow these steps to determine the region with the least standard deviation in level of interest based on the provided data.
To determine the region with the least standard deviation in level of interest, we need to analyze the histograms for each region. Standard deviation is a measure of how spread out the data is. The region with the least standard deviation will have data points that are more closely clustered around the mean (average) level of interest.
1. Examine the histograms for each region.
2. Look for the region where the data points are more tightly grouped around the mean level of interest. This indicates less variability and a smaller standard deviation.
3. Identify the region with the least standard deviation in level of interest.
To determine the region with the least standard deviation in the level of interest, examine the histograms for each region and identify the one where data points are tightly clustered around the mean level. Calculate the standard deviation for each region and select the region with the smallest value, indicating a more concentrated and less variable distribution of the level of interest.
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30 feet
50feet
Gilbert's family is building a walkway between the second floor balcony on his house and his tree house. How long
will the walkway be?
Q. How long is the walkway?
Answers
Answer:
what in the world am i
Step-by-step explanation:
juliet has 9 cups of orange syrup. she adds 32 fluid ounces of water to the syrup in order to prepare juice. find the total number of cups in the mixture.
Answers
Juliet will have 13 cups mixture of the orange juice and water .
Given :-
Juliet already has 9 cups of orange juice , and she adds 32 fluid ounces of water into the cup of orange uice syrup.
So, we have to find that how much cups of syrup does Juliet had ,
So, we know that
1 cup = 8 ounces
but she adds 32 ounces of water which means
= No. of ounces added / no. of ounces in a glass
= 32 / 8
= 4
therefore ,
9 + 4 = 13 glasses.
which means she has 13 glasses of the syrup made up of water and orange juice .
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Which of the following sets of numbers could repersent the three sides of a triangle?
a: 15,27,42
b:15,18,33
c:8,20,25
d:12,23,35
Answers
Answer:
c: 8,20,25
Step-by-step explanation:
You want to know which set of side lengths can form a triangle from the sets ...
a: 15,27,42b: 15,18,33c: 8,20,25d: 12,23,35Triangle inequalityA set of lengths can form a triangle if and only if the sum of the shorter two exceeds the longest.
In sets a, b, d, the sum of the shorter two is equal to the longest. These lengths will form a line segment, not a triangle.
A triangle can be formed by ...
c: 8,20,25 . . . . . . 8+20=28 > 25
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The number of members in an organization, M, changes as a function of t months. In which situation is M a linear function of t?
The membership doubles every year from the previous year.
1.The membership doubles every year from the previous year.
2.The membership increases by 1% every month from the previous month.
3.The membership decreases by 20 members every month from the previous month.
4.The membership increases by 50 members in each of the six warmest months and decreases by 50 members in each of the other six months.The membership doubles every year from the previous year.
Answers
Answer: D. The membership increases by 50 members in each of the six warmest months and decreases by 50 members in each of the other six months.
Step-by-step explanation: Because it is a linear function it needs to fluctuate so by using process of elimination D is the only answer that would look like a linear function by having that curved line we are looking for in a function.
I WILL CASH APP 15 DOLLARS TO ANYONE WHO CAN COMPLETE MY TWO VIRTUAL MATH TESTS FOR GEOMETRY!! PLEASE COMMENT OR RESPOND TO THIS!! I REALLY NEED HELP!!
Answers
Answer:
I will help
Step-by-step explanation:
HELP a map has a scale of 1 cm 35 feet use the map distance to find the actual distance
Answers
Answer:
Step-by-step explanation:
13. 35ft * 6cm = 210
14. 35ft * 3.2cm = 112
can someone help me with this math problem? thank you (The triangle below is equilateral. Find the length of side x in simplest radical form with a rational denominator.)
Answers
The length of side x, expressed in the simplest recent incarnation with a rational denominator, is √33 units in the given statement.
Provide a definition of a denominator?The denominator of a fractional is often the smaller number. It serves as an example of how something was divided into equal pieces. For example, if the portion was 3/4, 40 would be the fraction.
The altitude splits the side of an equilateral triangle in half.
So, the hypotenuse will be √11 + √11 = 2√11
From the Pythagoras theorem we have;
x² + (√11)² = (2√11)²
⇒ x = √(2√11)² - (√11)²
⇒ x = √33 units
Therefore, the length of x is √33 units.
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The length of x in simplest radical form with a rational denominator is (√33)/2.
Describe Triangle?A triangle is a three-sided polygon with three vertices and three angles. The sum of the interior angles of a triangle is always 180 degrees. Triangles are one of the basic shapes in geometry and are used in many different areas of mathematics, science, and engineering.
There are many different types of triangles, including equilateral triangles, isosceles triangles, and scalene triangles. An equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees. An isosceles triangle has two equal sides and two equal angles, while a scalene triangle has no equal sides or angles.
Let's call the length of the equilateral triangle's sides "s".
Since the median divides the equilateral triangle into two congruent right triangles, we can use the Pythagorean theorem to find x:
x² = s² - (s/2)²
x² = s² - s²/4
x² = 3s²/4
We're given that one of the sides of the equilateral triangle is √11, so we can use this to solve for s:
s² = (√11)²
s² = 11
Substituting this into the equation we found for x², we get:
x² = 3(11)/4
x² = 33/4
Taking the square root of both sides, we get:
x = √(33/4)
To simplify the radical, we can write:
x = √(33)/√4
x = (√33)/2
Therefore, the length of x in simplest radical form with a rational denominator is (√33)/2.
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A firm tested the durability of four different types of tires by having 4 Cadillacs each randomly equipped with one of the types of tires. In similar fashion, 4 Fords were used, each randomly equipped with one of the tire types. In a like manner, 4 Volkswagens were each randomly outfitted with one of the four tire brands. Then each of the twelve cars ran over a test track until the tires showed signs of serious wear. At that point the average mileage of the three cars using one type of tire was compared to the average mileage of the three cars using the second type of tires, and so on. This design is:
Answers
Answer:
Randomized block design
Step-by-step explanation:
In the scenario described above, the subjects which are the different car types are grouped into fixed units of 4 called blocks with each unit then assigned randomly to a particular treatment condition, which are the 4 different types of tire. In the end, the mean mileage for each block is taken and compared to that of the other blocks. This type of research design improves the reliability of the result obtained as it also eliminates the occurrence of systematic error in the course of our experiment.
How do you translate coordinates?
Answers
To translate an entire triangle, find the coordinates of each of the triangle's vertices, add the horizontal translation then plot the three translated points.
Whenever an object is moved from one location to the next without changing its size, shape, or orientation, a translation takes place. If we know which way and how far the figure to be moved, we can draw the translation in the coordinate plane.
Use P′(x+a,y+b) to translate the point P(x,y), a unit to the right, and b unit up. Use P′(x−a,y−b) to translate the point P(x,y), a unit to the left and b unit to the lower.
There are some steps of translating the triangle:
Step 1: Find the coordinates of each of the triangle's vertices in step one.
Step 2: Add the horizontal translation value to each vertex's x-coordinate and the vertical translation value to each point's y-coordinate to translate each point. If the horizontal translation is to the left or the vertical translation is downward for these translations, add a negative number.
Step 3: Plot the three translated points and draw the triangle by drawing a straight line between each pair of points.
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