Answers
Answer:
See below.
Step-by-step explanation:
I will assume that 3n is the last term.
First let n = k, then:
Sum ( k terms) = 7k^2 + 3k
Now, the sum of k+1 terms = 7k^2 + 3k + (k+1) th term
= 7k^2 + 3k + 14(k + 1) - 4
= 7k^2 + 17k + 10
Now 7(k + 1)^2 = 7k^2 +14 k + 7 so
7k^2 + 17k + 10
= 7(k + 1)^2 + 3k + 3
= 7(k + 1)^2 + 3(k + 1)
Which is the formula for the Sum of k terms with the k replaced by k + 1.
Therefore we can say if the sum formula is true for k terms then it is also true for (k + 1) terms.
But the formula is true for 1 term because 7(1)^2 + 3(1) = 10 .
So it must also be true for all subsequent( 2,3 etc) terms.
This completes the proof.
Inductive reasoning can include making generalizations based on specific instances, such as predicting that water freezes at 32°F based on past observations. The binomial theorem, a series expansion, is an example that can be proven using mathematical induction.
Explanation:Understanding Inductive Reasoning
Inductive reasoning is a method of reasoning in which the premises seek to supply strong evidence for the truth of the conclusion.
While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.
This form of reasoning allows us to create generalizations based on observations or patterns we have seen.
For instance, when we see that water freezes at 32° Fahrenheit, and we observe a temperature drop to 30°F at night, we can induce that there will be ice in the morning.
Mathematical Induction
Mathematical induction is a specific type of inductive reasoning that is used for proving that a statement is true for all natural numbers. It is a two-step process where we first prove that a base case is true, typically for n=1.
Then, we show that if the statement is true for an arbitrary positive integer n, this implies that it's also true for n+1.
This creates a domino effect, proving the initial statement for all natural numbers.
The binomial theorem is a classic example of series expansion that can be proven using mathematical induction.
The theorem expands powers of binomials in a form that involves summation of terms, and each term is a product of binomial coefficients and the variables raised to appropriate powers.
Series Expansions and Inductive Inferences
Series expansions in mathematics, such as the binomial theorem, are formulated based on patterns observed in specific instances.
These patterns are then generalized to hold for all numbers within the constraints of the theorem. In contrast, weak inductive inferences are those which cannot firmly establish a conclusion due to insufficient or irrelevant evidence.
Related Questions
tina’s preschool has a set of cardboard building blocks, each of which measures 9“ x 9“ x 4“. How many of these blocks will Tina need to build a wall 4 inches thick, 3 feet high, and 12 feet long?
Answers
4•9=36, and 36•9 is 324. 4•3 is 12, while 12•12 is 144.
324 - 144 = 180. I hope this helps!
Tina will need 64 cardboard building blocks to build the wall.
Step 1: Convert all measurements to inches.
The thickness of the wall is already 4 inches.
The height of the wall is 3 feet. Since 1 foot = 12 inches, then:
[tex]3 \text{ feet} = 3 \times 12 \text{ inches} = 36 \text{ inches}[/tex]
The length of the wall is 12 feet. Therefore:
[tex]12 \text{ feet} = 12 \times 12 \text{ inches} = 144 \text{ inches}[/tex]
Step 2: The volume of a rectangular wall can be found using the
formula:
[tex]\text{Volume} = \text{Length} \times \text{Height} \times \text{Thickness}[/tex]
Substituting the values:
[tex]\text{Volume} = 144 \text{ inches} \times 36 \text{ inches} \times 4 \text{ inches}[/tex]
[tex]\text{Volume} = 20736 \text{ cubic inches}[/tex]
Step 3: The dimensions of the block are 9 inches x 9 inches x 4 inches,
so:
[tex]\text{Volume}_{\text{block}} = 9 \text{ inches} \times 9 \text{ inches} \times 4 \text{ inches}[/tex]
[tex]\text{Volume}_{\text{block}} = 324 \text{ cubic inches}[/tex]
Step 4: To find out how many blocks are needed for the wall, divide the
total volume of the wall by the volume of one block:
[tex]\text{Number of blocks} = \frac{\text{Volume of wall}}{\text{Volume of one block}}[/tex]
[tex]\text{Number of blocks} = \frac{20736 \text{ cubic inches}}{324 \text{ cubic inches}}[/tex]
[tex]\text{Number of blocks} = 64[/tex]
find the volume of a sphere of radius 3 centimeters.
Answers
Answer:
36π cm³
Step-by-step explanation:
The volume (V) of a sphere is calculated using the formula
V = [tex]\frac{4}{3}[/tex]πr³ ← r is the radius = 3
V = [tex]\frac{4}{3}[/tex]π × 3³
= [tex]\frac{4}{3}[/tex]π × 27
= 4π × 9 ( cancelling the 3 and 27 )
= 36π cm³
Here is your answer
B) [tex]36×pi[/tex] [tex]{cm}^{3}[/tex]
REASON:
We know that,
Volume of sphere= [tex] 4/3 pi×{r}^{3} [/tex]
Here r= 3cm
So, V= [tex] 4/3 × pi× {3}^{3} cm^3 [/tex]
= [tex] 4×pi × {3}^{2} cm^3 [/tex]
= [tex] 4×9× pi cm^3 [/tex]
= [tex] 36× pi cm^3 [/tex]
HOPE IT IS USEFUL
What is the area of this triangle?
Answers
We use the vertical side as the base: it is a vertical segment, so it's length is the difference of the y coordinates of its endpoints:
[tex] b = y_2-y_1[/tex]
The height would be the segment starting from [tex](x_3,y_3)[/tex], perpendicular to the base. Since this is a horizontal segment, its length is the difference of the x coordinates of its endpoints:
[tex]h = x_3-x_1[/tex]
So, the area is given by
[tex]A = \dfrac{bh}{2} = \dfrac{(y_2-y_1)(x_3-x_1)}{2}[/tex]
Answer:
A = 18
Step-by-step explanation:
Drop a perpendicular from (x₃, y₃) to point A on the opposite side.
The vertical line containing A is the base of the triangle, and the horizontal line is its height,
The formula for the area of a triangle is
A = ½bh
b = y₂ - y₁ = 6
h = x₃ - x₁ = 6
A = ½ × 6 × 6 = 18
The area of the triangle is 18.
Krystal City has a population of 40,500 people. It’s population is increasing at a rate of 3.8% each year. Write a function that represents each population s a function of time.
Answers
Answer:
[tex]y=40,500(1.038^{x})[/tex]
Step-by-step explanation:
Let
x----> the time in years
y----> the population
we know that
[tex]100\%+3.8\%=103.8\%=103.8/100=1.038[/tex]
so
[tex]y=40,500(1.038^{x})[/tex]
Final answer:
For Krystal City's population of 40,500 increasing at 3.8% annually, the function is [tex]P(t) = 40500(1 + 0.038)^{t[/tex].
Explanation:
A population function can be represented as:
[tex]P(t) = P_0(1 + r)^{t[/tex]
Where:
P(t) is the population after time tP0 is the initial populationr is the growth rate per yearFor Krystal City with a population of 40,500 and a growth rate of 3.8% annually, the function would be:
[tex]P(t) = 40500(1 + 0.038)^{t[/tex]
A car went 120 miles in 3 hours. At this rate how long will it take for the car to travel in 350 miles
Answers
Answer:
8.75 hours
Step-by-step explanation:
120 miles in 3 hours = 40 mph
350 divided by 40 = 8.75
can u please help me solve 6x-3(2-3x)
Answers
Answer:
[tex]\boxed{\bold{15x-6}}[/tex]
Step By Step Explanation:
Expand [tex]\bold{-3\left(2-3x\right)}[/tex]
[tex]\bold{-6+9x}[/tex]
Rewrite Equation
[tex]\bold{6x-6+9x}[/tex]
Simplify [tex]\bold{6x-6+9x}[/tex]
[tex]\bold{15x-6}[/tex]
➤ [tex]\boxed{\bold{Mordancy}}[/tex]
When an elephant sat down to order a half of a third of a quarter of an eighty foot bun and a frankfurter, son was it longer than three feat o shorter
Answers
1/2 of 1/3 of 1/4 of 80ft
= 1/2 of 1/3 of (1/4 of 80ft)
= 1/2 of 1/3 of 20ft
= 1/2 of (1/3 of 20ft)
= 1/2 of (20/3) ft
= 20/6 ft
> 18/6 ft
= 3ft
So the order was longer than 3 feet.
Which equation when graphed has a maximum value at x = −2? A) y = −x2 − 20x − 16, B) y = −x2 − 16x − 12, C) y = −4x2 − 20x − 16, D) y = −4x2 − 16x − 12
Answers
Answer:
Option D. [tex]y=-4x^{2} -16x-12[/tex]
Step-by-step explanation:
using a graphing tool
Graph and determine the vertex in each case
we know that
If the equation has a maximum value at x=-2, then the x-coordinate of the vertex must be equal to -2 and the parabola open downward
case A) [tex]y=-x^{2} -20x-16[/tex]
The vertex is the point [tex](-10,84)[/tex]
case B) [tex]y=-x^{2} -16x-12[/tex]
The vertex is the point [tex](-8,52)[/tex]
case C) [tex]y=-4x^{2} -20x-16[/tex]
The vertex is the point [tex](-2.5,9)[/tex]
case D) [tex]y=-4x^{2} -16x-12[/tex]
The vertex is the point [tex](-2,4)[/tex] -------> is the answer
see the attached figure
The equation with a maximum value at x = -2 is D) y = -4x^2 - 16x - 12, determined by finding the vertex of each quadratic equation and identifying the one with the vertex x-coordinate at -2.
The given equations are all in quadratic form, which can have a maximum value at their vertex point if the leading coefficient is negative. To find the equation that has a maximum value at x = -2, we need to put the equations in vertex form, which is y = a(x - h)^2 + k, where (h, k) is the vertex. By transforming the given equations, we seek the equation that has its vertex at x = -2. This is done by completing the square or using the formula h = -b/(2a) to find the x-coordinate of the vertex.
Let's apply the formula h = -b/(2a) to each equation:
For A) y = -x^2 - 20x - 16, we have a = -1, b = -20, so h = -(-20)/(2 × -1) = 10, which is not equal to -2.For B) y = -x^2 - 16x - 12, using the same approach, we find h = -(-16)/(2 × -1) = 8, again not equal to -2.For C) y = -4x^2 - 20x - 16, with a = -4, b = -20, h = -(-20)/(2 × -4) = 2.5, which is not equal to -2.For D) y = -4x^2 - 16x - 12, we find h = -(-16)/(2 × -4) = -2, which matches the required vertex x-coordinate.Therefore, the correct equation with a maximum value at x = -2 is D) y = -4x^2 - 16x - 12.
Write a linear equation that goes through the points (0,4) and (4,10).
A. y =2⁄3x – 4
B. y =2⁄3x + 4
C. y =3⁄2x – 4
D. y =3⁄2x + 4
Answers
Answer:
D
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
To calculate m use the slope formula
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
with (x₁, y₁ ) = (0 , 4) and (x₂, y₂ ) = (4, 10)
m = [tex]\frac{10-4}{4-0}[/tex] = [tex]\frac{6}{4}[/tex] = [tex]\frac{3}{2}[/tex]
note the line passes through (0, 4) ⇒ c = 4
y = [tex]\frac{3}{2}[/tex] x + 4 ← equation of line → D
Be awesome! Is a new cross fit gym in town. The gym is almost set to its grand opening. The contractors are doing flooring. Gymnasium flooring cost $4.50 per square foot. The gym has 2 main rooms, one in a shape of a regular hexagon and the other in a shape of a regular nonagon. Both rooms have side lengths of 50 feet. Which of the following represents the closest estimate for the total of gymnasium flooring material? A: 29229 B: 40000
C: 69545 D: 98774
Answers
Answer:
D
Step-by-step explanation:
We need to find total area of the two rooms (hexagon and nonagon) and then multiply that by 4.5 to get total cost.
Area of Hexagon = [tex]\frac{3\sqrt{3}a^{2} }{2}[/tex]
Where a is the side length (which is 50)
Hence,
Area of Hexagon = [tex]\frac{3\sqrt{3}a^{2} }{2}=\frac{3\sqrt{3}(50)^{2} }{2}=6495.2[/tex]
Also, area of Nonagon is given by [tex]6.1818*s^2[/tex]
Where s is the length of the side
Hence,
Area of Nonagon = [tex]6.1818*s^2=6.1818*(50)^2=15,454.5[/tex]
Total floor area = 6495.2 + 15,454.5 = 21,949.7
Hence,
Total Cost = 21,949.7 * 4.5 = 98,773.65
THis is closest to the answer choice D.
What are the x and Y intercepts of Y=x^2+3x-10
Answers
Answer:
X-int: (2,0) , (-5,0)
Y-int: (0,-10)
Devon exercised the same amount of time each day for 5
days last week. His exercise included walking and swimming.
Each day he exercised, he walked for 10 minutes. He
exercised for a total of 225 minutes last week. What is the
number of minutes Devon swam each of the 5 days last
week? And show how!
Answers
Answer:
He spent 35 minutes swimming each day.
Step-by-step explanation:
10x5=50
The multiply by 5 is the numbers of days.
225-50=175
(The 50 minutes subtracted is the time Devon walked)
175 divided by 5 = 35
He spent 35 minutes swimming
Confirm/Check
35 times 5 =175
( Again, The multiply by 5 is the 5 days.)
175+50= 225
Therefore, He swam 35 minutes each day and walked 10 minutes each day.
He swam 175 minutes in total and walked 50 minutes in total
Hoped This Helped!
The number of minutes Devon swam each day is
How do we make equations from word problems?The word problems must be thoroughly read and understood. Then we can use the given data to make the equations.
We can find the number of minutes Devon spent swimming as follows:He exercised for a total of 225 minutes in the last week.
It is also given that he exercised the amount each day for 5 days.
Therefore, the total number of minutes of exercise per day = 225/5
= 45 minutes.
It is given that he spends 10 minutes everyday walking.
Therefore the total number of time Devon spend swimming = 45 - 10
= 35 minutes
Therefore, we have found that the number of minutes Devon swam each day is 35 minutes.
Learn more about word problems here: https://brainly.com/question/13818690
#SPJ2
what shapes are perpendicular sides only?
Answers
Answer:
Step-by-step explanation:
A square
Hannah and Anthony are siblings who have ages that are consecutive odd integers. The sum of their ages is 92. Which equations could be used to find Hannah’s age,h, if she is the older sibling?
Answers
Answer:
the answer is 47
Step-by-step explanation:
8(-2/7)(-1/2)
a -4/7
b 4/17
c 7/4
d 8/7
Answers
Answer:
d 8/7
Step-by-step explanation:
8 * -2 * -1
---- ----
7 2
Multiply the numerators
8*-2*-1 = 16
Multiply the denominators
7*2 =14
Put the numerator over the denominator
16/14
Both the numerator and the denominator can be divided by 2
8/7
Is there a rigid transformation that would map ΔABC to ΔDEC
Answers
Answer:
Yes. a rotation about point c
Step-by-step explanation:
just answered it.
A rigid transformation is a transformation that preserves the shape and size of an object. It includes translations, rotations, and reflections. Without specific information about the coordinates or measurements of the triangles, it is not possible to determine if a rigid transformation exists.
Explanation:A rigid transformation is a transformation that preserves the shape and size of an object. It includes translations, rotations, and reflections. In order for triangle ΔABC to be mapped to triangle ΔDEC, there must be a combination of translations, rotations, and reflections that can bring the two triangles into congruence.
However, without specific information about the coordinates or measurements of the vertices of the triangles, it is not possible to accurately determine if a rigid transformation exists that can map ΔABC to ΔDEC.
Order the numbers from least to greatest.
99, π2, 9.8
From least to greatest, the numbers are
Answers
Answer:
Step-by-step explanation:
Unfortunately, your π2 is improperly formed. I will assume that you actually meant π² ("pi squared").
Then the three numbers are 99, 9.87, 9.8.
Smallest: 9.8
Largest" 99 (did you mean 9.9?)
Under the assumptions I have made, the numbers, in increasing order, are 9.8, 9.87, 99
Answer:
From least to greatest, the numbers are [tex]99,\ \pi^2,\ 9.8[/tex]
Step-by-step explanation:
Given : The numbers are [tex]99,\ \pi^2,\ 9.8[/tex]
To find : Order the numbers from least to greatest ?
Solution :
Writing all numbers in same from,
We know, [tex]\pi=3.14[/tex]
[tex]\pi^2=(3.141)^2=9.86[/tex]
Numbers are [tex]99,\ 9.86,\ 9.8[/tex]
The least number is 9.8.
The greatest number is 99.
From least to greatest, the numbers are [tex]9.8,\ 9.86,\ 99[/tex]
or [tex]99,\ \pi^2,\ 9.8[/tex]
The period of a function is 4 pi how many cycles of the function occur in a horizontal length of 12 pi
Answers
Answer:
3 cycles
Step-by-step explanation:
in one period, you have one full cycle
because the period for this function is 4pi, you have one cycle per 4pi
if you have a total of 12pi, you would have 3 cycles (12pi/4pi=3)
hope this helps!
Answer:
1. 3
2. horizontal stretch
3. A.
Step-by-step explanation:
evaluate the expression -10-6+4 divided by (-0.5)(-2)
Answers
Answer:
-12
Step-by-step explanation:
numerator: -10 - 6 + 4 = -16 + 4 = - 12
denominator: (-0.5)(-2) = 1 Try this on your calculator.
Answer
-12 / 1 = - 12
Which of the following graphs represents the equation y + 2 = 3(x-1)?
Answers
Answer:
Graph C
Step-by-step explanation:
y + 2 = 3x - 3
y = 3x - 5
Graph C touches (0,-5) and no other graphs do.
Solve the system.
2/3 x - 1 /2 y = 1
1/ 4 x + 3/8 y = -1
A) (3, 2)
B) (9, 10)
C) ( 9/ 2 , 5)
D) (- 1 /3 , - 22 /9 )
Answers
Answer:
[tex]\large\boxed{D)\ \left(-\dfrac{1}{3},\ -\dfrac{22}{9}\right)}[/tex]
Step-by-step explanation:
[tex]\left\{\begin{array}{ccc}\dfrac{2}{3}x-\dfrac{1}{2}y=1&\text{multiply both sides by 6}\\\dfrac{1}{4}x+\dfrac{3}{8}y=-1&\text{multiply both sides by 8}\end{array}\right\\\left\{\begin{array}{ccc}6\!\!\!\!\diagup^2\cdot\dfrac{2}{3\!\!\!\!\diagup_1}x-6\!\!\!\!\diagup^3\cdot\dfrac{1}{2\!\!\!\!\diagup_1}y=6\cdot1\\8\!\!\!\!\diagup^2\cdot\dfrac{1}{4\!\!\!\!\diagup_1}x+8\!\!\!\!\diagup^1\cdot\dfrac{3}{8\!\!\!\!\diagup_1}y=8\cdot(-1)\end{array}\right[/tex]
[tex]\underline{+\left\{\begin{array}{ccc}4x-3y=6\\2x+3y=-8\end{array}\right}\qquad\text{add both saides of the equations}\\.\qquad6x=-2\qquad\text{divide both sides by 6}\\.\qquad x=-\dfrac{2}{6}\\\\.\qquad\boxed{x=-\dfrac{1}{3}}\\\\\text{Put the value of x to the second equation:}\\\\2\left(-\dfrac{1}{3}\right)+3y=-8\\\\-\dfrac{2}{3}+3y=-8\qquad\text{add}\ \dfrac{2}{3}\ \text{to both sides}\\\\3y=-\dfrac{24}{3}+\dfrac{2}{3}[/tex]
[tex]3y=-\dfrac{22}{3}\qquad\text{divide both sides by 3}\ /multiply\ both\ sides\ by\ \frac{1}{3}/\\\\\boxed{y=-\dfrac{22}{9}}[/tex]
X-intercept (__,__)
Y-intercept (__,__)
Answers
Look at the picture.
The x-intercepts are where the graph crosses the x-axis, and the y-intercepts are where the graph crosses the y-axis.
x-intercept (-250, 0)
y-intercept (0, 100)
Solve for x.
x2 - 2x = 0
A. 0,-2
B. 0,2
c. 1, -2
D. 1.2
Answers
Answer: B. 0, 2
Step-by-step explanation:
x² - 2x = 0
x(x - 2) = 0 Factored the left side
x = 0 and x - 2 = 0 Applied the Zero Product Property
x = 2 Solved the remaining equation
Therefore, x = 0 and x = 2
What is the value of w?
Answer options: 100, 75, 40, 52.5
Answers
➷ Opposite angles in a cyclic quadrilateral total to 180 degrees
180 - 80 = 100
w = 100 degrees
✽➶ Hope This Helps You!
➶ Good Luck (:
➶ Have A Great Day ^-^
↬ ʜᴀɴɴᴀʜ ♡
Write the slope intercept form of -y = 4-x
Answers
-y = 4 - x (Original)
Multiply by -1.
y = -4 + x
Reorder the terms.
y = x - 4
Please consider marking this answer as Brainliest to help me advance.
Slope intercept form:
y=mx+b
Right now we have:
-y=4-x
x=1 because it’s not being multiplied by anything
y-intercept = 4 because it’s what you’re adding
Then we just need to move 4 to the other side of “x”
And with that, we conclude with a line in slope intercept form:
-y=x-4
It’s a negative 4 and a positive x, because you’re supposed to do the opposite
If it’s a negative, you add. If it’s a positive, you subtract.
Jeff has recorded the cost of insurance premiums on his car for a particular insurance company.
Years Payments
1 $1,200
6 $7,200
9 $10,800
Daniel pays $1,150 per year for insurance premiums on his car. Whose unit rate is lower in terms of cost per year, and what is the rate?
A.
Daniel's unit rate is lower. It is $1,150 per year.
B.
Jeff's unit rate is lower. It is $1,200 per year.
C.
Daniel's unit rate is lower. It is $1,200 per year.
D.
Jeff's unit rate is lower. It is $1,150 per year.
E.
Jeff and Daniel have the same unit rate. It is $1,150 per year.
Answers
For 1 year Jeff pays $1200.
For 6 years he pays 7200 / 6 = 1200 per year.
For 9 years he pays 10800 / 9 = 1200 per year.
Jeff's unit rate is $1,200 per year.
Daniel pays $1,150 per year.
150 is less than 1200, so the answer would be:
A. Daniel's unit rate is lower. It is $1,150 per year.
Answer:
View the picture for your answer!
Have a great day.
PLEASE PLEASE HELP ME ASAP
Answers
The correct answer is 36.
Will mark brainliest! Show work for 5 and 6!❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️
Answers
Answer:
6. decrease / 19.68 ,5. increase / 4.4
Step-by-step explanation:
What is the phase shift of a periodic function?
Answers
Answer and Explanation :
To find : What is the phase shift of a periodic function?
Solution :
The Phase shift is defined as a horizontal shift in a function in any direction.
Horizontal stretches will change the period of the function and that horizontal shift is called a phase shift or the amount of shift in a wave horizontally.
The phase shift is measured in degrees.
Refer the attached figure below for the pictorial representation of teh phase shift.
In the figure, [tex]\frac{\pi }{2} \text{ to } \pi[/tex] is showing the phase shift.
Example - Taking a general example of sin function
[tex]y = A \sin(B(x + C)) + D[/tex]
In this the phase shift is at C (positive is to the left).
find a factor pair of 80 that has a sum of 21
Answers
16 and 5 are the factors
Answer: 5 and 16
Step-by-step explanation:
Because 5 plus 16 is 21.
then 5 times 16 is 80