In the image below, DE || BC. Find the measure of EC. Set up a proportion and solve for the measure. Show your work and label your answer. Please help me -- I will mark brainliest; If your response is inapropriate, or just a way to get points, it will get reported! Thank you so much !

Answer:

B. This is an example of causation because the application of fertilizer caused the plant to improve its leaf color.

Answer:

B

Step-by-step explanation:

Answer:

Part a: The transformation matrix is the clockwise rotation matrix of π/4.

Part b: The hyperplane would move towards the mean of class 1.

Part c: The distance will remain in the Euclidean Space  due to the rotation transformation only.

Step-by-step explanation:

As the complete question is not available, the question is searched online and a reference question is obtained which has 3 parts as follows:

Part a:

The decision boundary  after transformation coincides with the line x1 = x2, the two class means must lie on a line that is normal to the decision boundary, i.e. on x1 = −x2. This implies that the transformation matrix Γ is a clockwise rotation transformation of π/4, given as

                                               [tex]\Gamma=\left[\begin{array}{cc}\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\\\frac{-\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{array}\right][/tex]

Part b:

If the prior probability of class 0 was increased after transformation, then the decision boundary of BDR would still have the same normal as before,  i.e.,[tex]w=(1/\sqrt{2},-/\sqrt{2})^T[/tex], but move toward the mean of class 1.

Part c:

Noting that

[tex]||\bold{T}_x-\bold{T}_y||^2=(x-y)^T \bold{T}^T\bold{T}(x-y)\\||\bold{T}_x-\bold{T}_y||^2=(x-y)^T(x-y)\\||\bold{T}_x-\bold{T}_y||^2=||x-y||^2[/tex]

This indicates that the distance is still the same and is in Euclidean space. This is due to the fact that rotation transformations does not affect the distances between the points.

Answer:

No, it is not.

Step-by-step explanation:

The set [tex] C = \mathbb{N} \times \mathbb{N}[/tex] contains every ordered pair of Natural numbers, while B only contains those pairs in which both values in each entry are the same. Therefore, C is a bigger set than B, but B is not equal to C because for example C contains [tex][1] \times [2] [/tex] and B doesnt because 1 is not equal to 2.

Answer:

(a) 2 feet.

(b) 2 feet.

Step-by-step explanation:

We have been given that the velocity function [tex]v(t)=\frac{1}{\sqrt{t}}[/tex] in feet per second, is given for a particle moving along a straight line.

(a) We are asked to find the displacement over the interval [tex]1\leq t\leq 4[/tex].

Since velocity is derivative of position function , so to find the displacement (position shift) from the velocity function, we need to integrate the velocity function.

[tex]\int\limits^b_a {v(t)} \, dt[/tex]

[tex]\int\limits^4_1 {\frac{1}{\sqrt{t}}} \, dt[/tex]

[tex]\int\limits^4_1 {\frac{1}{t^{\frac{1}{2}}} \, dt[/tex]

[tex]\int\limits^4_1 t^{-\frac{1}{2}} \, dt[/tex]

Using power rule, we will get:

[tex]\left[\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}}\right] ^4_1[/tex]

[tex]\left[\frac{t^{\frac{1}{2}}}{\frac{1}{2}}}\right] ^4_1[/tex]

[tex]\left[2t^{\frac{1}{2}}\right] ^4_1[/tex]  

[tex]2(4)^{\frac{1}{2}}-2(1)^{\frac{1}{2}}=2(2)-2=4-2=2[/tex]

Therefore, the total displacement on the interval  [tex]1\leq t\leq 4[/tex] would be 2 feet.

(b). For distance we need to integrate the absolute value of the velocity function.

[tex]\int\limits^b_a |{v(t)|} \, dt[/tex]

[tex]\int\limits^4_1 |{\frac{1}{\sqrt{t}}}| \, dt[/tex]

Since square root is not defined for negative numbers, so our integral would be [tex]\int\limits^4_1 {\frac{1}{\sqrt{t}}} \, dt[/tex].

We already figured out that the value of [tex]\int\limits^4_1 {\frac{1}{\sqrt{t}}} \, dt[/tex] is 2 feet, therefore, the total distance over the interval [tex]1\leq t\leq 4[/tex] would be 2 feet.

Answer:

1/2

Step-by-step explanation:

(1/4)^x

when x=(1/2)

(1/4)^(1/2)

this can be rewritten in this way

√(1/4)

2 is the radical

then we distribute the root

√1 / √4

and solve

1/2

Answer:

9

Step-by-step explanation:

Number of tulips per row has to be a factor of the number of available tulips.

Red tulips: 45 = 3×3×5

Yellow tulips: 63 = 3×3×7

Highest no. of tulips each row is 9 (3×3 = 9)

Answer:Circumference = 2 \pi r

circumference = 2 \pi (1.5)

circumference = 9.42 feet

The circumference is 9.42 feet, so we need to get fencing for 10 feet (assuming the fence comes in feet, i.e. you cant buy half a foot)

1 feet = $.075

10 feet = $0.75 x 10 = $7.50

Answer:

  65°

Step-by-step explanation:

Since the high is given, it is convenient to use that value with a cosine function to model the temperature. The function will be ...

  T = A + Bcos(C(x-D))

where A is the average temperature, B is the difference between the high and the average, C is π/12, reflecting the 24-hour period, and D is the time at which the temperature is a maximum. "x" is hours after midnight.

We have chosen to use a 24-hour clock with x=16 at 4 pm. Then the value of T at 9 am is ...

  T = 70 +20cos((π/12(9 - 16)) = 70 +20cos(7π/12) ≈ 64.824

The temperature at 9 am is about 65°.

Final answer:

To find the temperature at 9 AM, a sinusoidal function is constructed with an amplitude of 20 degrees, a vertical shift of 70 degrees, and a phase shift adjusted for a high at 4 PM. Plugging in the time value of 9 AM into the function, it is determined that the temperature at 9 AM is approximately the same as the average or midline temperature, which is 70 degrees.

Explanation:

To find the temperature at 9 AM using a sinusoidal function, we first identify essential characteristics of the function. The maximum temperature (high) of 90 degrees occurs at 4 PM (which we'll take as 16 hours on a 24-hour clock) and the average temperature for the day is 70 degrees, which is also the midline of the sinusoidal function. Since this is a typical daily temperature cycle, we assume that the minimum temperature occurs 12 hours after the maximum temperature. Thus, the temperature will have a periodicity of 24 hours. The amplitude of the temperature variation will be the difference between the high temperature and the average temperature (which is 20 degrees in this case).

The sinusoidal function can be written in the form:

T(t) = A · sin(B(t - C)) + D

Where:

C is the phase shift (16 hours for 4 PM)

Using this information, the sinusoidal function for temperature throughout the day is:

T(t) = 20 · sin((2π/24)(t - 16)) + 70

We then plug in t = 9 (for 9 AM), and calculate the temperature:

T(9) ≈ 20 · sin((2π/24)(9 - 16)) + 70

Which gives us, to the nearest degree:

Temperature at 9 AM ≈ 70 degrees.

Since the value of the sine function ranges from -1 to 1, at 9 AM (7 hours before the maximum temperature at 4 PM), the sine value would be negative indicating that the temperature is rising from the minimum towards the average. However, due to the characteristics of sine, the temperature at 9 AM will actually be the same as the temperature at 19 hours (7 PM), which is also 70 degrees.

Answer:  B. [tex]\dfrac{1}{98}[/tex].

Step-by-step explanation:

Given : A hat contains four red marbles, two blue marbles, seven green marbles, and one orange marble.

Total marbles = 4+2+7+1 = 14

P(Blue)= [tex]\dfrac{2}{14}=\dfrac{1}{7}[/tex]

P(Orange) = [tex]\dfrac{1}{14}[/tex]

if we randomly select two marbles , then the probability of selecting  one will be orange and one will be blue marble =  P(Blue) x P(Orange)  [both events are independent]

[tex]=\dfrac{1}{7}\times\dfrac{1}{14}=\dfrac{1}{98}[/tex]

Hence, the probability that one will be orange and one will be blue is  [tex]\dfrac{1}{98}[/tex].

Therefore , the correct answer is  B. [tex]\dfrac{1}{98}[/tex].

Answer:

$151,800

Step-by-step explanation:

For the publisher, the expected revenue is $26.40 per copy. For 225,000 copies, the revenue is [tex]225000\times26.40 = 5,940,000[/tex]

The expenses incurred by the publisher are as follows:

Cost of 1 print = $3.75

Cost of 225,000 prints = [tex]225000\times3.75 = 843,750[/tex]

Editing/Design = $27,500

Publicity/Advertising/Administrative = $135,150

Author's fee = 6.5% of retail price per copy for 225,000 copies = [tex]225000\times26.40\times6.5/100= 386,100[/tex]

Total cost = 843,750 + 27,500 + 135,150 + 386,100 = $1,392,500

Let the cost of kelp for copies be k.

Then the total cost = 1,392,500 + k

If the expected profit is 4,092,100, then

Revenue = total cost + profit

5,940,000 = 1,392,500 + k + 4,092,100

5,940,000 = 5,484,600 + k

k = 5,940,000 - 5,484,600 = 455,400

Since Carlita is paying [tex]\frac{1}{3}[/tex] of k, her share of the kelp =

[tex]\frac{1}{3}\times455400 = 151800[/tex]

Carlita will pay $151,800

Answer:

-4.99

1.9435

3.25E4

1.56e-9

Step-by-step explanation:

Answer:    [tex]\dfrac{12}{13}[/tex]

Step-by-step explanation:

Let A = he known the answer then A' = he guess the answer.

B = he answered it correctly

As per given , we have

[tex]P(A)=\dfrac{3}{4}\ \ ,\ \ P(A')=\dfrac{1}{4}[/tex]

[tex]P(B|A)=1[/tex]

[tex]P(B|A')=\dfrac{1}{4}[/tex]

By Bayes theorem , we have

[tex]P(A|B)=\dfrac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}\\\\ P(A|B)=\dfrac{1\times\dfrac{3}{4}}{1\times\dfrac{3}{4}+\dfrac{1}{4}\times\dfrac{1}{4}}\\\\= \dfrac{12}{13}[/tex]

The probability that the student knows the answer given that he answered it correctly is  [tex]\dfrac{12}{13}[/tex] .

In 15 coin tosses, an unbiased coin can land tails either exactly 8 times or exactly 5 times in 46,683 ways.

The question relates to the concept of probability, specifically calculating the number of outcomes in coin tosses. The outcomes in which a coin lands tails exactly 8 times or exactly 5 times can be calculated using the formula for combinations, which is nCr = n! / r!(n-r)!. For n=15 (the number of trials) and r=8 (the number of successful outcomes), the number of ways the coin can land tails 8 times is 15C8 = 15! / 8!(15-8)! = 43,680 ways. Similarly, for the coin to land tails 5 times the number of ways is 15C5 = 15! / 5!(15-5)! = 3,003 ways. Therefore, the coin can land tails either exactly 8 times or exactly 5 times in 43,680 + 3,003 = 46,683 ways.

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The correct answer is that there are [tex]\(\binom{15}{8} + \binom{15}{5}\)[/tex] ways for the coin to land tails either exactly 8 times or exactly 5 times.

To solve this problem, we will use the concept of combinations, which is denoted by the binomial coefficient [tex]\(\binom{n}{k}\)[/tex], where [tex]\(n\)[/tex] is the total number of trials and [tex]\(k\)[/tex] is the number of successful trials. The binomial coefficient calculates the number of ways to choose [tex]\(k\)[/tex] successes from [tex]\(n\)[/tex] trials.

For the first part of the question, where we want the coin to land tails exactly 8 times out of 15 tosses, we calculate the number of combinations using the binomial coefficient:

[tex]\[ \text{Number of ways for 8 tails} = \binom{15}{8} \][/tex]

For the second part of the question, where we want the coin to land tails exactly 5 times out of 15 tosses, we calculate the number of combinations similarly:

[tex]\[ \text{Number of ways for 5 tails} = \binom{15}{5} \][/tex]

To find the total number of ways for either event to occur, we add the two separate probabilities together:

[tex]\[ \text{Total number of ways} = \binom{15}{8} + \binom{15}{5} \][/tex]

Now we calculate each binomial coefficient:

[tex]\[ \binom{15}{8} = \frac{15!}{8!(15-8)!} = \frac{15!}{8!7!} \][/tex]

[tex]\[ \binom{15}{5} = \frac{15!}{5!(15-5)!} = \frac{15!}{5!10!} \][/tex]

We can simplify these expressions by canceling out common factors in the numerator and the denominator:

[tex]\[ \binom{15}{8} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \][/tex]

[tex]\[ \binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} \][/tex]

After calculating these values, we would add them together to get the final answer. However, since we are not providing numerical calculations here, we leave the answer in the form of the sum of the two binomial coefficients:

[tex]\[ \text{Total number of ways} = \binom{15}{8} + \binom{15}{5} \][/tex]

This is the number of ways the coin can land tails either exactly 8 times or exactly 5 times in 15 tosses.

Answer:

Step-by-step explanation:

2,2+1,2+1+3,2+1+3+5,...

2,3,6,11,18,27

Answer:

  L = s^2/(30.25Cd)

Step-by-step explanation:

In accident investigation, the speed of a vehicle can be estimated using a polynomial function that relates speed (s) to the length of skid marks (L). The drag coefficient Cd will depend on the condition of the road surface and tires, but might be expected to be between 0.7 and 0.8.

If the skid marks end in a collision, the length of the marks that might have been made can be estimated using this formula, then that length added to the actual length of marks to estimate the original speed. The speed at the point of collision can be estimated by the damage caused, and/or the movement created.

In the above formula, length is in feet, and speed is in miles per hour.

In Physics, a polynomial function such as[tex]y = -gt^2 + v0*t + h0[/tex] can model an object's motion under gravity. A rational function like P = a / (1 + bQ) can model the relationship between supply and demand in Economics.

In the field of Physics, polynomial functions are often used to model physical phenomena. For example, the motion of an object under the force of gravity can be represented by a second-degree polynomial, or quadratic function. The equation[tex]y = -gt^2 + v0*t + h0[/tex]is an example of a quadratic function modeling free fall, where g represents the acceleration due to gravity, v0 is initial velocity, t represents time and h0 is initial height.

On the other hand, rational functions are used in various fields too. In Economics for instance, it can model situations like the relationship between supply and demand in market equilibrium. A simple model could be P = a / (1 + bQ) where P represents the price, Q is the quantity of good sold, and a, b are constants.

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Answer:

B) ∫₂⁵ ∜x dx

Step-by-step explanation:

The factor is 3/n, so b − a = 3

The expression under the radical is 2 + 3k/n, so a = 2.  Therefore, b = 5.

The function is f(x) = ∜x.

Plugging into a definite integral:

∫₂⁵ ∜x dx

Answer:

√820

√36/83

√48/81

√111

√17

Step-by-step explanation:

Wszystkie liczby, które nie są wymierne, są uważane za irracjonalne. Liczbę niewymierną można zapisać jako liczbę dziesiętną, ale nie jako ułamek. Liczba niewymierna ma niekończące się powtarzające się cyfry po prawej stronie przecinka dziesiętnego.

mam nadzieję, że to pomoże

Answer:

  4·6 = 24 ≠ 12 = LCM(4, 6)

Step-by-step explanation:

Any pair of numbers with a common factor will refute Sarah's conjecture.

One such pair is 4 and 6, which have a product of 24. The LCM of 4 and 6 is 12, which is that product divided by their common factor of 2.

Sarah's statement is incorrect.

Sarah's statement that the least common multiple (LCM) of any two whole numbers is obtained by merely multiplying them together is incorrect. For example, consider the whole numbers 8 and 12. The product of these two numbers is 8  * 12 = 96. However, the LCM of 8 and 12 is actually 24, since 24 is the smallest number that is divisible by both 8 and 12. To find the LCM of two numbers, you need to list their multiples and find the smallest number that appears on both lists (in this case, the multiples of 8 are 8, 16, 24, 32, ... and the multiples of 12 are 12, 24, 36, 48, ...).

Answer:

3

Step-by-step explanation:

Answer:

Claire traveled for 9 days.

Step-by-step explanation:

Given:

Total Distance traveled = 701 miles

Distance traveled each day = 80 miles

Distance traveled on last day = 61 miles

We need to find the number of days Claire traveled.

Solution:

Let the number of days Claire traveled be denoted by 'd'.

Now we can say that;

Total Distance traveled is equal to sum of Distance traveled each day multiplied by number of days and Distance traveled on last day.

framing in equation form we get;

[tex]80d+61=701[/tex]

Now Subtracting both side by 61 using Subtraction Property of Equality we get;

[tex]80d+61-61=701-61\\\\80d = 640[/tex]

Now Dividing both side by 80 we get;

[tex]\frac{80d}{80}=\frac{640}{80}\\\\d=8[/tex]

Hence Claire traveled 80 miles in 8 days and 61 miles on last day making of total 9 days of travel.

Answer:

13 square units

Step-by-step explanation:

You could calculate the length of each side using Pythagorean theorem, then use Heron's formula to find the area.  But there's an easier way: find the area of the rectangle that contains the triangle, and subtract the areas of the smaller triangles in the corners.

The area of the rectangle is 4 × 7 = 28.

The area of the upper left triangle is ½(2)(4) = 4.

The area of the upper right triangle is ½(3)(5) = 7.5.

The area of the lower right triangle is ½(1)(7) = 3.5.

So the area of the triangle is:

28 − 4 − 7.5 − 3.5 = 13

Answer: The initial number of shoes in the store is 300.

Step-by-step explanation:

Given : Kelly is a salesperson at a shoe store, where she must sell a pre-set number of pairs of shoes each month.

At the end of each work day the number of pairs of shoes that she has left to sell that month is given by the equation

[tex]S=300-15x[/tex] , where S =  Number of pair of shoes Kelly still needs to sell.

x =  Number of days she has worked that month.

When x= 0 , we get S= 300

i.e. When she started working in that month , she has 300 pairs of shoes to sell.

Therefore , Number 300 means that the initial number of shoes in the store is 300

Answer:

Definition: Two sets are equal if they contain the

same elements. I.e., sets A and B are equal if

∀x[x ∈ A ↔ x ∈ B].

Notation: A = B.

Recall: Sets are unordered and we do not distinguish

between repeated elements. So:

{1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.

Definition: A set A is a subset of set B, denoted

A ⊆ B, if every element x of A is also an element of B.

That is, A ⊆ B if ∀x(x ∈ A → x ∈ B).

Example: Z ⊆ R.

{1, 2} ⊆ {1, 2, 3, 4}

Notation: If set A is not a subset of B, we write A 6⊆ B.

Example: {1, 2} 6⊆ {1, 3}

Answer:

Step-by-step explanation:

Answer:

The probability that the selected person is a woman engineering major is 0.0396.

Step-by-step explanation:

The proportion of students at the university who are women is 0.42.

P (W) = 0.42

The proportion of students at the university who are engineering majors is 0.18.

P (E) = 0.18

The proportion of engineering majors that are women is 0.22.

P (W|E) = 0.22

The proportion of students at the university that are woman and engineering major is:

[tex]P (W|E)=\frac{P(W\cap E)}{P(E)} \\P(W\cap E)=P(W|E)\times P(E)\\= 0.18\times0.22\\=0.0396[/tex]

Thus, the probability that the selected person is a woman engineering major is 0.0396.

The probability that a randomly selected student from the university is a woman engineering major is 99/100.

To find this probability, we will use the information given about the percentages of women and engineering majors at the university, as well as the percentage of women among the engineering majors.

First, let's denote the total number of students at the university as T.

According to the information given:

- 42% of the students are women, so the number of women students is 0.42T

- 18% of the students are engineering majors, so the number of engineering students is 0.18T

- Of the engineers, 22% are women, so the number of women engineering students is 0.22 \times 0.18T.

Now, we want to find the probability that a randomly selected student is a woman engineering major. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. In this case, the favorable outcomes are the number of women engineering majors, and the total number of possible outcomes is the total number of students.

The probability P that a randomly selected student is a woman engineering major is:

[tex]\[ P = \frac{\text{Number of women engineering majors}}{\text{Total number of students}} \][/tex]

[tex]\[ P = \frac{0.22 \times 0.18T}{T} \][/tex]

Since [tex]$T$[/tex]is in both the numerator and the denominator, it cancels out, leaving us with:

[tex]\[ P = 0.22 \times 0.18 \][/tex]

[tex]\[ P = 0.0396 \][/tex]

To express this probability as a fraction, we can write it as:

[tex]\[ P = \frac{396}{10000} \][/tex]

Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4, we get:

[tex]\[ P = \frac{99}{2500} \][/tex]

Answer:

the answer is g21T

Step-by-step explanation:

a3z, b6Y, C9X, d12W, e15V, f18U next step in the series is E. g24t

A mathematical series consists of a pattern in which the next term is obtained by adding the two terms in-front. An example of the Fibonacci number series is: 0, 1, 1, 2, 3, 5, 8, 13, … For instance, the third term of this series is calculated as 0+1+1=2.

1) Arithmetic Sequences.

2) Geometric Sequences.

3) Exponent Sequences.

4) Two-Stage Sequences.

5) Mixed Sequences.

6) Alternating Sequences.

7) Fibonacci Sequences.

8) A Combination of Sequences' Types.

In mathematics, a series is the cumulative sum of a given sequence of terms. Typically, these terms are real or complex numbers, but much more generality is possible

You can find the right answer in number series by taking the difference between consecutive pairs of numbers, which form a logical series. In this example, the differences between succeeding pairs of numbers are 1,2, 4, 8, 16, and 32. So, the next difference must be 64.

To learn more about series of the number , refer

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Answer:

3,604,389

Step-by-step explanation:

Monthly payment P = [tex]\frac{a}{[(1+r)^{n} - 1] / [r(1+r)^{n}] }[/tex]

where a = total loan amount, r = periodic rate, n = number of payment periods

a = 500,000 ; r = 0.12 ; n = (5 x 12) = 60 months

P = [tex]\frac{500000}{[(1+0.12)^{60} - 1] / [0.12 (1+0.12)^{60} ] }[/tex]

P = [tex]\frac{500000}{(896.597) / (107.712)}[/tex]

p = 60,073.151

Total amount paid = 60073.151 x 60 = 3,604,389

Final answer:

Liza will have to pay a monthly amount of P2,997.75 for a loan of P500,000 at a 12% interest rate compounded monthly over 5 years. The total amount paid at the end of the term will be P179,865.

Explanation:

To determine the total sum Liza will pay for a loan of P500,000 with an interest rate of 12% compounded monthly over 5 years, we first need to calculate the monthly payment she would need to make. The problem is essentially asking to find the annuity payment for a present value, which can be calculated using the formula for present value of an ordinary annuity:

PV = PMT * [1 - (1 + r)-n] / r

Where:

PV = Present Value of the annuity (P500,000 in this case)

PMT = Monthly payment

r = Monthly interest rate (12% per year compounded monthly, so 0.12/12 per month)

n = Total number of payments (5 years × 12 months/year = 60 payments)

First, we calculate the monthly interest rate:

r = 0.12 / 12 = 0.01 or 1%

Now, setting up the equation:

500,000 = PMT * [1 - (1 + 0.01)-60] / 0.01

We find that PMT = 500,000 / 166.7916 (Using the present value factor derived from the reference information)

PMT = P2997.752

Therefore, the monthly payment Liza has to make is P2,997.75. To find the total sum paid after 5 years:

Total Sum = Monthly Payment * Number of Payments

Total Sum = P2,997.75 * 60 = P179,865

Answer:

3

Step-by-step explanation:

You'll need to use the Angle Bisector Theorem.

CU / ZU = BC / BZ

From the first part of the problem, you used Pythagorean theorem to find that BZ = 10.

Let's say that CU = x.  That means ZU = 8 − x.  Plugging in values:

x / (8 − x) = 6 / 10

Cross multiply:

10x = 6(8 − x)

Solve:

10x = 48 − 6x

16x = 48

x = 3

Answer:

Step-by-step explanation:

Let us assume that we have n different dots on a paper.  We are to connect pairwise by a line.  We have to find out how many lines can be formed.

Let us prove by induction.

If there is one dot then we have no line = 1(1-1) =0

Thus n(n-1) is true for 1 dot

Let us assume that for n dots we have n(n-1) lines

Add one more point now total points are n+1.

Already the existing n points are connected by a line.

So the extra point has to be connected to each of n point

i.e. n lines should be added from the new point to the n points and again n lines from the points to the new point(Assuming lines are different if initial and final point are different)

So 2n lines would be added

So total number of lines for n+1 points

[tex]= n(n-1) +n+n= n^2-n+2n \\= n^2+n\\=n(n+1)[/tex]

Thus true for n+1 if true for n.  Already true for n =1

So proved by induction for all natural numbers n.

Using mathematical induction, we prove the number of lines connecting pairs of dots for n dots is n(n-1)/2. We confirm the base case for n=1 and then assume the proposition holds for k, leading to it holding for k+1, thus proving it for all natural numbers n.

The question involves calculating the number of lines that can be drawn to connect each pair of dots on a paper. To find this number for n dots, we can use mathematical induction.

Let's denote P(n) as the proposition that for n dots, the number of lines required to connect each pair of dots is n(n-1)/2, which simplifies to the sum of the first n-1 natural numbers.

Base Case

For n=1, there are no lines needed since there is only one dot, and no connections to be made: P(1) holds because 1(1-1)/2 = 0.

Inductive Step

We assume that P(k) is true for some natural number k, meaning that k dots require k(k-1)/2 lines. Now, we consider n=k+1 dots. The new dot must be connected to each of the k existing dots, adding k new lines. Therefore, the total number of lines for k+1 dots is k(k-1)/2 + k, which simplifies to (k+1)k/2, and is exactly P(k+1).

By the principle of mathematical induction, since we have proven the base case and the inductive step, we conclude that P(n) holds for all natural numbers n.