Can anyone explain how to do mathematical induction. The question is below. Brainliest awarded for the right answer. Thanks.

Can Anyone Explain How To Do Mathematical Induction. The Question Is Below. Brainliest Awarded For The

Answers

Answer 1

The sum of the triangular numbers is [tex]\frac{n(n+1)(n+2)}{6}[/tex]

According to the question, the series of triangular numbers is given as in the form of the number of dots constituting the equilateral triangles i.e.
1, 3, 6, 10, 15 . . . . .n

what is triangular numbers?

triangular number are the sequence and series of the numbers and each number represents and constitute in visualization of  series of  equilateral triangle.

given series in the figure
1, 3, 6, 10, 15, . . . . . .  ., n
each number represents the number of dots containing in triangle
now the series can be given by
[tex]S = \frac{n(n+1)}{2}[/tex]  for n = 1, 2, 3, 4,. . . . .n
now , according to the question the sum of the series can be given as
⇒  ∑S
⇒   ∑[tex]\frac{n^2+n}{2}[/tex]
⇒  [tex]\frac{1}{2}[\sum n^2+\sum n]\\\frac{1}{2}[\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}] \\\frac{n(n+1)}{4}[\frac{(2n+1)}{3}+1] \\\frac{n(n+1)}{4}[\frac{(2n+1+3)}{3}] \\\frac{n(n+1)}{4}[\frac{(2n+4)}{3}] \\\frac{n(n+1)}{4}2[\frac{(n+2)}{3}]\\\frac{n(n+1)(n+2)}{6}[/tex]

Thus, the sum of the triangular number is given by [tex]\frac{n(n+1)(n+2)}{6}[/tex]

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Related Questions

Two gymnasts are running toward each other in a floor routine, and they plan to precisely time a flip to stay synchronized for the audience. The path of the gymnasts is parabolic and modeled by the following equations, where y is the height of the flip and x is the time in seconds:

Answers

Answer:

4 secs

Step-by-step explanation:

Since they have to be synchronized, their parabolic equations must be equal to one another

3(t^2 + 9 - 6t) - (t^2 + 25 - 10t) - t + 2 = 0

2t^2 - 9t + 4

(2t - 1)(t - 4) = 0

t = 1/2, 4

From the options, the answer is 4

The gymnasts will be at the same height during their flips at two different times: 1/2 seconds and 4 seconds.

so, correct option is: C.

Here, we have,

To determine when the gymnasts will be at the same height during their flips, we need to find the time (x) at which the equations for y are equal.

The given equations are:

y = –(x – 5)² + 3

y = –3(x – 3)² + x + 1

Setting the two equations equal, we have:

–(x – 5)² + 3 = –3(x – 3)² + x + 1

Expanding the squared terms:

–(x² – 10x + 25) + 3 = –3(x² – 6x + 9) + x + 1

Simplifying the equation:

–x² + 10x – 25 + 3 = –3x² + 18x – 27 + x + 1

Combining like terms:

–x² + 10x – 22 = –3x² + 19x – 26

Rearranging the equation:

2x² - 9x + 4 = 0

To solve this quadratic equation, we can factor it:

(2x - 1)(x - 4) = 0

Setting each factor equal to zero:

2x - 1 = 0 or x - 4 = 0

Solving for x:

2x = 1 or x = 4

Dividing by 2:

x = 1/2 or x = 4

Therefore, the gymnasts will be at the same height during their flips at two different times: 1/2 seconds and 4 seconds.

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Here's a graph of a linear function. Write the
equation that describes that function.
Express it in slope-intercept form

Answers

Answer:

y = 3/4x + 2

Step-by-step explanation:

let me know if you want an explanation :))

Alex, jas and stef each get a student loan to help with living expenses. they decide to allocate two fifths of their loans for food, and one-sixth for travel. what fraction of their student loan will be left to spend?

Answers

The fraction of their student loan will be left to spend is half of their loan.

A number expressed quotient, in which numerator is divided by denominator is called a fraction.

Let the loan amount be $100.

Expenses for food is given that two fifths of their loan

i.e.

= 2/5 × 100

= $40

Now,

Remaining part will be = $100- $40 = $60

So,

Expenses for travel are given that one sixth of their loan,

i.e.

= 1/6 × 60

= $10

So,

Remaining part will be = $60 - $10 = $50

Hence, the fraction of their student loan will be left to spend is half of their loan.

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HELP NOW PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE PLEASE

Answers

Answer:

-3/2

Step-by-step explanation:

First, you divide by taking the reciprocal of -2/5 which is -5/2.

Second, you get the answer which is -5/4.

Add that with -1/4 and you will get -6/4.

Simplify to get the answer which is -3/2.

Answer:

[tex]-\dfrac{3}{2}[/tex]

Step-by-step explanation:

Given expression:

[tex]\dfrac{\dfrac{1}{2}}{-\dfrac{2}{5}}+\left(-\dfrac{1}{4}\right)[/tex]

To divide fractions, flip the second fraction (make the numerator the denominator, and the denominator the numerator) then multiply it by the first fraction:

[tex]\implies \dfrac{1}{2} \times -\dfrac{5}{2}+\left(-\dfrac{1}{4}\right)[/tex]

To multiply the fractions, multiply the numerators and multiply the denominators:

[tex]\implies \dfrac{1 \times (-5)}{2 \times 2} +\left(-\dfrac{1}{4}\right)[/tex]

[tex]\implies -\dfrac{5}{4} +\left(-\dfrac{1}{4}\right)[/tex]

Apply the rule -a + (-b) = -a - b :

[tex]\implies -\dfrac{5}{4} -\dfrac{1}{4}[/tex]

As the denominators are the same, subtract the numerators and put the answer over the same denominator:

[tex]\implies \dfrac{-5-1}{4}[/tex]

[tex]\implies -\dfrac{6}{4}[/tex]

Simplify by dividing the numerator and denominator by the highest common factor:

[tex]\implies -\dfrac{6 \div 2}{4 \div 2}[/tex]

[tex]\implies -\dfrac{3}{2}[/tex]

True or false: The factor by which a row operation changed the determinant is equal to the determinant of the elementary matrix corresponding to that row operation.

Answers

The given statement is false.

What is the effect of a row operation on a determinant?

The factor by which a row operation intends to change the determinant is not equal to the determinant of the elementary matrix corresponding to that row operation. Rather, when a row is scaled up by a factor in a matrix, the determinant of that matrix also scales up by that factor.

Similarly, the factor by which a row operation changes the determinant is equal to the factor times the determinant of the elementary matrix corresponding to that row operation.

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A dilation maps (9, 12) to (3, 4). Find the coordinates of the point (8, 4) under the same dilation

Answers

Assuming the dilation is centered at the origin, the scale factor is 1/3.

This means the coordinates of the point (8,4) are [tex]\boxed{\left(\frac{8}{3}, \frac{4}{3} \right)}[/tex]

1. A bus traveled for 4 hours and 30 minutes to travel 183 miles. What was it's average speed?
A. 42.56 mph
B. 40.67 mph
C. 32.09 mph
D. 30.67 mph

Answers

Answer:

B or 40.67mph

Step-by-step explanation:

To solve we will use formula distance over total time, so are distance is 183miles and are time is 4 hours and 30 minutes. If we put that in hours we get 4.5 hours. Now we simply divide 183/4.5 to get about 40.67 .

Solve for x using quadratic formula :
abx^2 + (b^2 - ac)x - bc = 0​

Answers

Answer:

[tex]\boxed{\sf x = \dfrac{c}{b} \quad or \quad \dfrac{-b}{a}}[/tex]

Explanation:

Given expression: (ab)x^2 + (b^2 - ac)x + (-bc) = 0

Here given:

a = abb = b² - acc = -bc

Apply quadratic formula:

[tex]\sf x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{2a} \quad when \ ax^2 + bx + c = 0[/tex]

Insert values:

[tex]\sf x = \dfrac{-(b^2 - ac) \pm \sqrt{(b^2 -ac)^2-4(ab)(-bc)} }{2(ab)}[/tex]

[tex]\sf x = \dfrac{-b^2 + ac \pm \sqrt{\left(b^2-ac\right)^2+4abbc} }{2ab}[/tex]

[tex]\sf x = \dfrac{-b^2 + ac \pm \sqrt{b^4+2b^2ac+a^2c^2} }{2ab}[/tex]

[tex]\sf x = \dfrac{-b^2 + ac \pm \sqrt{\left(b^2+ac\right)^2} }{2ab}[/tex]

[tex]\sf x = \dfrac{-b^2 + ac \pm( b^2+ac )}{2ab}[/tex]

[tex]\sf x = \dfrac{-b^2 + ac +( b^2+ac )}{2ab} \quad or \quad \dfrac{-b^2 + ac -( b^2+ac )}{2ab}[/tex]

[tex]\sf x = \dfrac{2ac}{2ab} \quad or \quad \dfrac{-2b^2}{2ab}[/tex]

[tex]\sf x = \dfrac{c}{b} \quad or \quad \dfrac{-b}{a}[/tex]

(ab)x^2 + (b^2 - ac)x -bc= 0

a = abb = b² - acc = -bc

Apply quadratic formula:

[tex]\sf x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{2a} [/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{-(b^2 - ac) \pm \sqrt{(b^2 -ac)^2-4(ab)(-bc)} }{2(ab)}[/tex]

[tex]\\ \sf\Rrightarrow x= \dfrac{-b^2 + ac \pm \sqrt{\left(b^2-ac\right)^2+4abbc} }{2ab}[/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{-b^2 + ac \pm \sqrt{b^4+2b^2ac+a^2c^2} }{2ab}[/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{-b^2 + ac \pm \sqrt{\left(b^2+ac\right)^2} }{2ab}[/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{-b^2 + ac \pm( b^2+ac )}{2ab}[/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{-b^2 + ac +( b^2+ac )}{2ab} \quad or \quad \dfrac{-b^2 + ac -( b^2+ac )}{2ab}[/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{2ac}{2ab} \quad or \quad \dfrac{-2b^2}{2ab}[/tex]

[tex]\\ \sf\Rrightarrow x = \dfrac{c}{b} \quad or \quad \dfrac{-b}{a}[/tex]

Hi! So, I know I got this answer wrong, but I wasn't sure how to solve an equation with signs like: [brackets] in it, I've included my problem as an example, but can someone please teach me what those brackets mean, and how do I go about solving an equation (using this problem as an example) with brackets like these?

Answers

Step-by-step explanation:

Brackets is the bigger version of parentheses, you first solve the questions inside the parentheses, then move onto brackets.

For example, this question:

[tex]x=-1\\y=-2\\z=3[/tex]

[tex]5x-y[7-4(z-y)][/tex]

plug in x, y, and z.

[tex]5(-1)-(-2)[7-4(3-(-2))][/tex]
[tex]5(-1)-(-2)[7-4(5)][/tex]

[tex]5(-1)-(-2)[7-20][/tex]

[tex]5(-1)-(-2)[-13][/tex]

[tex]-5-(-2)[-13][/tex]

[tex]-5-26[/tex]

[tex]-31[/tex]

I hope you understand better now.

The graph of a system of inequalities shown

Answers

ya you got it right -8 can be represented as “b” and is the y-intercept, was that your question?

What is the area of this triangle?


1375 ft²


1500 ft²


3300 ft²


4500 ft²

Answers

Answer:

Area of a triangle = 1/2 * base * height

here ,

base = 120 ft

height = 25 ft

Area = 1/2* 120*25

= 60 * 25

= 1500 ft²

Answer 2

Answer:

1/2 * base * height

so 1/2 * 25 * 120

= 1500

Desiree is given an aptitude test with 50 multiple-choice questions. For every correct answer, Desiree will get 3 points. For every wrong answer, 1 point will be deducted. For every question unanswered, 0.5 point is deducted. Desiree did not leave any question unanswered and gets 110 points on the test.
If x is the number of questions Desiree answered correctly, then the equation that represents the given situation is
and the equation will have
.

Answers

Total number of multiple-choice questions that Desiree has to answer in Aptitude test = 50

Points given for every correct Answer = +3

Points deducted for every Incorrect Answer = -1

For every question unanswered ,

points Deducted = -0.5

Total Points Obtained by Desiree after Answering all the questions = 110

Number of Answers that Desiree answered correctly = x questions

Number of Incorrect Answers = (50 - x) questions

Then,the Equation representing above situation

→ 3 × x + ( -1 ) × ( 50 - x ) = 110

⇒3x - 50 + x = 110 ----------- equation that represents the given situation

⇒ 4x - 50 = 110

Adding 50, on both sides

→ 4x - 50 + 50 = 110 + 50

⇒ 4x = 160

Dividing both sides by, 4 we get

x = 40

Number of correct answers given by Desiree= 40 questions

Number of Incorrect Answers = 50 - 40

= 10 Questions .

Answer:

Desiree is given an aptitude test with 50 multiple-choice questions. For every correct answer, Desiree will get 3 points. For every wrong answer, 1 point will be deducted. For every question unanswered, 0.5 points is deducted. Desiree did not leave any questions unanswered and gets 110 points on the test.

If x is the number of questions Desiree answered correctly, then the equation that represents the given situation is

3(50 - x) + x = 110

and the equation will have

no solution

.

Step-by-step explanation:

PQ = 15cm and QR = 17cm. Calculate the perimeter of PQR.

Answers

can u please tell if the figure is a right angled triangle??...cuz in order to find the perimeter, Pythagoras theorem must be applied but it's only for right angled triangle...so specify please

Using a standard deck of cards, find the probability of selecting a jack, replacing the card, and then selecting a king.

please explain

Answers

The probability of selecting a jack, replacing the card, and then selecting a king is 1/169

How to determine the probability?

In a standard deck of cards, we have:

Total = 52

Jack = 4

King = 4

The probability of each is:

P(Jack) = 4/52

P(King) = 4/52

So, we have:

P = 4/52 * 4/52

Simplify

P = 1/13 * 1/13

Evaluate

P = 1/169

Hence, the probability is 1/169

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If m varies directly with n , and m = 8 when n = 24 . what is the value of n when m = 12

Answers

Answer:

the answer will be 36

Step-by-step explanation:

if you divide 24 by 8 it will be 3 so then you multiply 12 times 3 and it's 36

(c) (i) A new truck costs $15 000 and loses 23% of its value each year. Calculate the value of the truck after three years. ( c ) ( i ) A new truck costs $ 15 000 and loses 23 % of its value each year . Calculate the value of the truck after three years .​

Answers

Answer:

$6847.955

Explanation:

Use the compound interest formula, but the value decreases over time.

[tex]\sf A = P(1 - \dfrac{r}{100} )^t[/tex]

where 'A' is final amount, r is rate, t is time

Inserting P = $15,000, r = 23, t = 3 years

[tex]\sf A = 15000(1- \dfrac{23}{100} )^3[/tex]

[tex]\sf A = 6847.995[/tex]

Hence the value of truck after three years will be $6847.955.

A coordinate plane with 2 lines. The first line is labeled y equals f(x) and passes through (0, 4) and (1, 1) The second line is labeled y - g(x) and is horizontal passing through (negative 2, 2), (0, 2), and (2, 2). The lines intersect at a point that is slightly to the right of (0.5, 2).
If f(x) = −3x + 4 and g(x) = 2, solve for the value of x for which f(x) = g(x) is true.

Answers

The value of x if both lines intersect is 2/3

What is an equation?

The equation in mathematics is the relationship between the variables and the number and establishes the relationship between the two or more variables.

Given the equation of a line that passed through the points (0, 4) and (1, 1) expressed as h(x) = -3x +4

The equation of the second line g(x) passing through the points (-2, 2) and (0, 2) is g(x) = 2

If the lines intersect, then h(x) = g(x)

Substituting the functions to get the value of "x"

-3x + 4 = 2

Subtract 4 from both sides

-3x + 4 - 4 = 2 - 4

-3x = - 2

Divide both sides by - 3

x = 2 / 3

Hence the value of x if both lines intersect is 2/3

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Answer:

2/3

Step-by-step explanation:

edge23

Suppose you know the slope of a linear relationship and one of the points that it’s graph passes through how can you determine if the relationship is proportional or not?

Answers

Answer:

I exactly don't have or know the answer

Answer is
Use the graph and the given point to determine the second point. Connect the two points by a straight line. If the graph passes through the origin, the relationship is proportional and if the graph does not pass through the origin, the relationship is non-proportional.

what is the moop3fngonoi9w

Answers

Hbubvr vunnigvvtfryyfsgg

If salt (5.99 × 10–6 mol) is dissolved in 1.50 × 10–2 l of water, which expression can be used to find the molarity of the resulting solution? 2.50 × 10-8 m 2.50 × 103 m 3.99 × 10–4 m 3.99 × 104 m

Answers

The molarity of the resulting solution is 3.99 × 10⁻⁴ M. The correct option is the third option 3.99 × 10⁻⁴ M

Molarity of a solution

From the question, we are to determine the molarity of the resulting solution

From the given information,

Number of moles = 5.99 × 10⁻⁶ mol

Volume = 1.50 × 10⁻² L

Using the formula,

Molarity = Number moles / Volume

∴ Molarity = (5.99 × 10⁻⁶) / (1.50 × 10⁻²)
Molarity = 3.99 × 10⁻⁴ M

Hence, the molarity of the resulting solution is 3.99 × 10⁻⁴ M. The correct option is the third option 3.99 × 10⁻⁴ M

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Answer:

The answer is 3.99 × 10–4 M

Step-by-step explanation:

I just did the assignment on Edge :)

Missing angles/math please help guys

Answers

Answer:

Step-by-step explanation:

Deon has bought 18 pounds of dog food. He feeds his dog
Write your answer in simplest form.
2|3
pounds for each meal. For how many meals will the food last?

Answers

Answer:

6 to 9 meals

Step-by-step explanation:

9 If he feeds 2 pounds per meal. 18÷2 =9

6 if he feeds 3 pounds per meal. 18÷3=6

Need Help Fast!!!!!! The graph of the piecewise function f(x) is shown. f(x) What is the range of f(x)?

Answers

Answer:

The second option

Step-by-step explanation:

If you look at the graph, it appears that from negative infinity to 0, the line is just constant, so the range of that would simply be the constant value or in this case 4. from 0 to infinity it appears the line is decreasing at a constant rate and should go towards negative infinity as x goes towards infinity. So the range would be -infinity < f(x) <= 4

The table shows some information about the lifetimes,
t
, in hours, of some lightbulbs.

Lifetime Frequency
25 <
t
≤ 50 70
50 <
t
≤ 100 76
100 <
t
≤ 150 29
150 <
t
≤ 200 91
200 <
t
≤ 250 0
250 <
t
≤ 300 15
300 <
t
≤ 350 23

Estimate the mean lifetime of a bulb.
Give your answer rounded to 2 DP.

Answers

Answer:

25 < t < 350

Step-by-step explanation:

The mean lifetime of a bulb ranges from 25 to 350 hrs but the question is unclear.

solve X-2=7 answer the question pls pls​

Answers

Answer:

x = 9

Step-by-step explanation:

x-2 = 7

you solve by adding 2 to both sides to maintain equality. and you add by 2 to cancel out the -2 to isolate the x.

x = 9

Answer: 9
if you remove the X with 9 it would be 9-2=7

Which currency is it

Answers

Answer:

pounds (£)

Step-by-step explanation:

pounds are used in countries such as the UK and South Georgia

How many square feet of outdoor carpet will we need for this hole.

Answers

Step-by-step explanation:

area of rectangle= 12×6

=72ft^2

A2=9ft^2 whereby the base of the hole is 3ft

height is 6ft

therefore the square is 18 ft

(-4,3)

-5-4-3

24

5

432

-14

-2

-3-

A

(0,1)

1 2

(4,-1)

Which linear function is represented by the graph?

Of(x) = -2x + 1

Of(x)=x+1

O f(x) = x+1

Of(x)=2x+1

Answers

Answer: [tex]f(x)=-\frac{1}{2}x+1[/tex]

Step-by-step explanation:

The slope is [tex]\frac{-1-1}{4-0}=-\frac{1}{2}[/tex], which matches the second option.

Linear function that is represented by the graph is f(x)= -1/2 x+1.

Here, we have,

From the graph,

y - intercept of the linear function is 1 , i.e. c = 1

and there are two points on the line (-4, 3), (4, -1)

We can get the equation of line by applying slope-intercept formula,

Slope of the line,

m = y₂ -y₁ / x₂-x₁

so, we get,

m = -1 -3 / 4 + 4

   = -4/8

   = -1/2

Now applying slope-intercept formula,

y = mx+c

Putting the values,

=> y = -1/2 x + 1

=>f(x) = -1/2 x + 1.

Therefore, linear function that is represented by the graph is

f(x) = -1/2 x + 1.

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how many sets of 5 students can be selected out of 30 students?

Answers

Answer:

142 506

Step-by-step explanation:

here the order does not matter

Then

we the number of sets is equal to the number of combinations.

Using the formula :

the number of sets is 30C5

[tex]C{}^{5}_{30}=\frac{30!}{5!\left( 30-5\right) !}[/tex]

      [tex]=142506[/tex]

There are 142506 ways in which 5 students can be selected out of 30 students.

How can a certain number of individuals be selected using a combination?

The selection of 5 students out of 30 students can be achieved with the use of combination since the order of selection is not required to be put into consideration.

By using the formula:

[tex]\mathbf{^nC_r = \dfrac{n!}{r!(n-r)!}}[/tex]

where;

n = total number of individual in the set = 30r = number of chosing individuals to be selected = 5

[tex]\mathbf{^nC_r = \dfrac{30!}{5!(30-5)!}}[/tex]

[tex]\mathbf{^nC_r = \dfrac{30!}{5!(25)!}}[/tex]

[tex]\mathbf{^nC_r = 142506}[/tex]

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Need help with this question (pic included)

Answers

Answer:

answer= 3

hope this helps if not I'll try to do it again

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