There are probably other ways to approach this, but I'll focus on a proof by induction.
The base case is that n = 1. Plugging this into the expression gets us
n^3+11n = 1^3+11(1) = 1+11 = 12
which is a multiple of 6. So that takes care of the base case.
----------------------------------
Now for the inductive step, which is often a tricky thing to grasp if you're not used to it. I recommend keeping at practice to get better familiar with these types of proofs.
The idea is this: assume that k^3+11k is a multiple of 6 for some integer k > 1
Based on that assumption, we need to prove that (k+1)^3+11(k+1) is also a multiple of 6. Note how I've replaced every k with k+1. This is the next value up after k.
If we can show that the (k+1)th case works, based on the assumption, then we've effectively wrapped up the inductive proof. Think of it like a chain of dominoes. One knocks over the other to take care of every case (aka every positive integer n)
-----------------------------------
Let's do a bit of algebra to say
(k+1)^3+11(k+1)
(k^3+3k^2+3k+1) + 11(k+1)
k^3+3k^2+3k+1+11k+11
(k^3+11k) + (3k^2+3k+12)
(k^3+11k) + 3(k^2+k+4)
At this point, we have the k^3+11k as the first group while we have 3(k^2+k+4) as the second group. We already know that k^3+11k is a multiple of 6, so we don't need to worry about it. We just need to show that 3(k^2+k+4) is also a multiple of 6. This means we need to show k^2+k+4 is a multiple of 2, i.e. it's even.
------------------------------------
If k is even, then k = 2m for some integer m
That means k^2+k+4 = (2m)^2+(2m)+4 = 4m^2+2m+4 = 2(m^2+m+2)
We can see that if k is even, then k^2+k+4 is also even.
If k is odd, then k = 2m+1 and
k^2+k+4 = (2m+1)^2+(2m+1)+4 = 4m^2+4m+1+2m+1+4 = 2(2m^2+3m+3)
That shows k^2+k+4 is even when k is odd.
-------------------------------------
In short, the last section shows that k^2+k+4 is always even for any integer
That then points to 3(k^2+k+4) being a multiple of 6
Which then further points to (k^3+11k) + 3(k^2+k+4) being a multiple of 6
It's a lot of work, but we've shown that (k+1)^3+11(k+1) is a multiple of 6 based on the assumption that k^3+11k is a multiple of 6.
This concludes the inductive step and overall the proof is done by this point.
A football player attempts to kick a football across a field. The path of the football can be modeled by the function h(x) = − 1/225 x^2 + 2/3 x, where x is the horizontal distance from the kick, and h(x) is the height of the football above the ground, when both are measured in feet. How far from the kicker will the football land?
PLEASE I NEED THIS ASAP MY TEST HAS AN ANSWERR PLEASEEE
What is the value of x?
-3x + 8 = 38
Answer:
-10
hope it helps
Pls help me so I don’t get left back.
Answer:
a) ∠ IHJ is 55° because MIJ is a straight angle
∠IHN is 125° because vertically opposite from ∠IHM
∠ NHM is 55° because it is vertically opposite from ∠ IHJ
b) Angles 4 and 6 are interior angles. They equal to 180. A similar example to that is ∠5 and ∠3.
c)Angle 1 and 5 are corresponding angles. They equal the same. Anothe example would be ∠6 and ∠2.
Hope this helped!
Need help with math earn 10 points
Answer:
Step-by-step explanation:
60
Help please test is due in 16 mins i will make brainlest
Answer: the answer is 118
Step-by-step explanation:
x=5,y=6
4(52)+(3)(6)
4(52)+(3)(6)
=118
Answer:
45^2 + 36 = 2061
Step-by-step explanation:
ASAP! MATH WIZZES PLEASE HELP!! ALGEBRA 1
Factor the following trinomials:
1) x^2 + 10x + 16
2) x^2 + 21x + 20
3) x^2 - 6x + 8
Step-by-step explanation:
1) (x+2)(x+8)
2) (x+1)(x+20)
3) (x-2)(x-4)
Answer:
1) (x+2)(x+8)
2) (x+1)(x+20)
3) (x-2)(x-4)
Step-by-step explanation:
A prism has a length of 5 meters, height of 6 meters, and width of 1 meter.
The length of the prism is
m.
The width of the prism is
m.
The height of the prism is
m.
unit cubes fill the prism.
The volume of the prism is
m3.
Answer: 30 m3 (meters cubed)
Step-by-step explanation:
To get the volume of a rectangular prism, multiply the width times the length times the height.
1 x 5 x 6 = 30
Add the unit label
I have a lot of questions I need to do for final so yeah can you guys estimate 16×22×9. Estimate 4×26×15. Estimate 6×21×19. Estimate 39×95×10. Estimate 52×8×12. Estimate 16×8×19. Estimate 29×21×2. Estimate 32×21×7. Estimate 12×72×9. Estimate 8×19×78. Estimate 20×48×12. Please if you can do this I own you my life.
Step-by-step explanation:
16×22×9 is abt 3168. 4×26×15 is abt 1560. 6×21×19 is abt 2394. 39×95×10 is abt 37050. 52×8×12 is abt 4992. 16×8÷19 is abt 2432. 29×21×2 is abt 1218. 32×21×7 is abt 4704. 12×72×9 is abt 7776. 8×19× 78 is abt 11856. 20×48×12 is abt 11520
Can someone answer this please
Answer:
sorry i cant...
Step-by-step explanation:
Answer: 40
Step-by-step explanation: so the area of the triangle is 150 the area of the garden is 50 so he wants them in the triangle not the garden so 150 - 50 is 100 and each bag of rocks will cover 2.5 so 1-- divided by 2.5 is 40
If the perimeter of a triangle is 31 ft and two of the side lengths are 15 ft and 10 ft, what is the length of the third side?
A. 56 ft
B.25 ft
C.16 ft
D.6 ft
Answer:
D would be the answer
Step-by-step explanation:
31 - 15 = 16
16 - 10 = 6
converting fractions to decimals 18/20
Answer:
The Answer is 0.9!!!!!!!!!!!!!
Answer:
0.9
Step-by-step explanation:
Converting the fraction 18/20 to a decimal is 0.9
1st Method: Simplify
18/20 can be simplified to 9/10. 9/10 is equivalent to 0.9
2nd Method: Use long division
Convert the fraction to a decimal by dividing the numerator by the denominator.
im very confused on this please help me:)
The constant rate of change is how much y increases every time x increases by 1. We might be confused because in this table, we see a number of points, but we don't see any consecutive x-values, so it might seem difficult. But no worries!
All you have to do is calculate using the formula, Δy divided by Δx. What does Δy divided by Δx mean?
All we have to do is look at two points. Let's take, for example, (-2, -10) and (2, -8).
-2 and 2 are the x-values of the points and -10 and -8 are the y-values of the points as coordinates are in the format of (x, y).
All we have to do is minus the y-coordinates and then proceed the minus the x-coordinates. The order does not matter, as in you can do either of these ways:
[(-10) - (-8)] ÷ [(-2) - (2)] = -2 / -4 = 1/2
or
[(-8) - (-10)] ÷ [(2) - (-2)] = 2 / 4 = 1/2
So the constant rate of change or the answer is 1/2.
help pleasee!!!!!!!!
Answer:
14.2 ounce bottle
Step-by-step explanation:
the 14.2 ounce bottle Is 2 cents cheaper per ounce
what angle is complementary to 1 ?
Answer:
angle 2
Step-by-step explanation:
angle 1 and 2 together make 90 degrees
If the cross section of a triangular prism is parallel to its base, what shape will the cross section be?
Answer:
It will be a hexagon
Step-by-step explanation:
Triangle JKL is rotated 180 about the origin, resulting in triangle J’K’L. What are the coordinates of the vertices of J’K’L? Move the coordinates into the tae to indicate vertices of triangle J’K’L’.
HELP ME PLZ
The verticies of J’K’L would be, J'(-a, -b), K'(-c, -d), and L'(-e, -f).
What are transformations?Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.
Dilation: The preimage is scaled up or down to create the image.
Reflection: The picture is a preimage that has been reversed.
Rotation: Around a given point, the preimage is rotated to create the final image.
Translation: The image is translated and moved a fixed amount from the preimage.
Given, Triangle JKL is rotated 180 about the origin, resulting in triangle J’K’L.
Assuming the coordinates of J(a, b), K(c, d) and L(e, f).
We know a rotation of 180° to (x , y) ⇒ (- x, - y).
Therefore, The vertices of J’K’L would be, J'(-a, -b), K'(-c, -d), and L'(-e, -f).
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How many lines of symmetry are in a regular polygon?
seven
twice as many as the number of sides
half as many as the number of sides
the same number as the number of sides
Answer:
the same number as the number of sides
lines of symmetry are in a regular polygon is the same number as the number of sides.
the correct option is (D)
What is Line of symmetry?A line of symmetry is a line that cuts a shape exactly in half. This means that if you were to fold the shape along the line, both halves would match exactly.
The number of lines of symmetry in a regular polygon is equal to the number of sides.
For example,
It has four lines of symmetry and four sides. A regular pentagon has 5 sides and 5 lines of symmetry.
Hence, lines of symmetry in a regular polygon is equal to the number of sides.
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A mobile phone plan has a base fee of $50 per month. The monthly cost increases by $10 for every gigabyte of data used. Let c represent the monthly cost and g the gigabytes of data used.
what is the price of you pizza with all the toppings?
Each answer is under each Part
1. What is the price of you pizza with all the toppings?
$16
The pizza costs 12 dollars, and each topping costs 1.75 dollars, there are 4 toppings, so it's 12 + 1.75*4 = 12 + 4 = 16 dollars.
2. How much will you pay in tax?
$0.80
It costs 16 dollars and tax is 5%, so 5% of 16. 16*5% = 16*0.05 = 0.8, so 0.8 dollars, or $0.80
3. What is the total cost including tax?
$16.80
Add the cost of the pizza with the tax, 16 + 0.8 = 16.8, $16.80
4. How much will each of you need to pay to cover the total price?
$4.20
There are 4 people total, You and 3 friends, so divide the total cost by 4, 16.80/4 = 4.2, so $4.20
Answer:
0.95
Step-by-step explanation:
A bicycle normally sells for $239.99. It is now on sale for 25% off. An employee Baron is able to save an extra 10% off the sale price. How much, to the nearest dollar, would Baron need to pay for his bicycle.
Answer:
$162.00
Step-by-step explanation:
Start by finding the price after 25% off.
239.99 x .75 = 179.9925
Then, an extra 10%.
179.9925 x .90 = 161.99325
Rounded to the nearest dollar is 162.
Help me with these two questions ASAP (I will try to mark brainliest)
Answer:
1. C
2.B
IG
Step-by-step explanation: CALCULATE
Answer:
the answer would be a is correct
If an impulse travels 100 m/s, about how long will it take the impulse to travel 10 meters?
Answer:
0.1 seconds
Step-by-step explanation:
10/100=0.1
Answer:
1/10th of a second or .1seconds
Step-by-step explanation:
100m/s meaning its traveling 100 meters a second.
Since its going to travel 10 meters, it will take 1/10 the time since 10/100 = 1/10
The time is in seconds so it will take them 1/10th of a second.
How many faces have an area of 6 cm2?
Answer:
2
Step-by-step explanation:
There are 6 sides:
3 by 4 = 3*4 = 12, no
3 by 3 = 3*3 = 9, no
5 by 3 = 5*3 = 15, no
Triangles:
Half of 3 by 4 = 1/2 * 3* 4 = 1/2 * 12 = 6, yes
The other triangle is the same, so 2
Answer:
two
Step-by-step explanation:
Area of the rectangle on the left = 4 * 3 = 12 cm^2
area of the square = 3^2 = 9 cm^2
Area of the rectangle on the right = 5 * 3 = 15 cm^2
Area of the two triangles = (4 * 3)/2 = 6 cm^2
Round 72,934 to the nearest hundreds place and enter the value below.
Answer:
72,900
Step-by-step explanation:
9 is in the hundreds place so therefore you should look to the left which us 3 and 3 is less than 5 so the answer is 72,900
Find the volume of this prism.
The volume of the prism is 420 in³.
Option C is the correct answer.
We have,
From the figure,
Length = 7 in
Width = 5 in
Height = 12 in
Now,
To find the volume of the prism, we can use the formula:
Volume = Length * Width * Height
Substituting these values into the formula:
Volume = 7 * 5 * 12
Simplifying the expression:
Volume = 420 cubic inches
Thus,
The volume of the prism is 420 in³.
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Help with math please earn 10 points
Answer:
Option 3: y = -0.5 + 27
Step-by-step explanation:
Where the + 27 comes from: It comes from where the line starts
Negative slope always goes down, and has negatives. Hence the answer is 3.
A football player attempts to kick a football across a field. The path of the football can be modeled by the function h(x) = − 1/225 x^2 + 2/3 x, where x is the horizontal distance from the kick, and h(x) is the height of the football above the ground, when both are measured in feet. How far from the kicker will the football land?
Answer:
Cristiano Ronaldo
Step-by-step explanation:
because he has a lot of shot power
HELP MEEEEEEEEEEEE
Katherine randomly those one candy from a jar and put it back in the jar. She repeated this process many times. The table shows the number of
times she chose each color of candy.
Color of Candy Number of Times
Red :12
Green :11
Yellow :12
Purple :8
Blue: 7
Based on the data in the table, what is the probability that the next piece of candy Katherine pulls out of the jar will be blue?
IF YOU ANSWER CORRECTLY I WILL GIVE BRAINIEST!,
2. At a fair, each person can spin two wheels of chance. The first wheel has the numbers 1, 2, and 3. The second wheel has the letters A and B. (d) List all the possible outcomes of the compound event. (e) If you spin both wheels, what is the probability that you get either a 1 or an A? Explain.
And pls don’t answer if you don’t know it.
Answer:
Possible outcomes: 1a 1b 2a 2b 3a 3b
Outcomes for spinning a one = 2
1a, and 1b 2/6
Outcomes for spinning an "a" = 3
1a, 2a, and 3a.
1a is already in the first set so we can exclude it. 2/6
2/6+2/6=4/6
There is a 4/6 or 2/3 chance of getting either an a or an 1
Answer:
(d) 1A, 1B, 2A, 2B, 3A, 3B
(e) 2/3 or 66.66% chance
Explanation is below
Step-by-step explanation:
(d) The Total outcomes are 3*2 = 6, because there are 3 numbers and 2 letters.
The outcomes are:
1A
1B
2A
2B
3A
3B
(e) Of the 6 outcomes, 3 of them have A and 2 of them have 1. But one of them has an A and 1, so it is 4 outcomes: 1A, 1B, 2A, and 3A.
So it is 4/6 = 2/3, or 66.66...% chance