Proof by induction
Base case:
n=1: 1*2*3=6 is obviously divisible by six.
Assumption: For every n>1 n(n+1)(n+2) is divisible by 6.
For n+1:
(n+1)(n+2)(n+3)=
(n(n+1)(n+2)+3(n+1)(n+2))
We have assumed that n(n+1)(n+2) is divisble by 6.
We now only need to prove that 3(n+1)(n+2) is divisible by 6.
If 3(n+1)(n+2) is divisible by 6, then (n+1)(n+2) must be divisible by 2.
The "cool" part about this proof.
Since n is a natural number greater than 1 we can say the following:
If n is an odd number, then n+1 is even, then n+1 is divisible by 2 thus (n+1)(n+2) is divisible by 2,so we have proved what we wanted.
If n is an even number" then n+2 is even, then n+1 is divisible by 2 thus (n+1)(n+2) is divisible by 2,so we have proved what we wanted.
Therefore by using the method of mathematical induction we proved that for every natural number n, n(n+1)(n+2) is divisible by 6. QED.
whats 48 divided by 2
Answer:
My answer to the question is 24.
What is the area of the figure?
Answer:
33[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
Calculate the area of the large figure by multiplying the length and the width (7[tex]\frac{1}{4}[/tex]*6). Then, calculate the area of the smaller figure (5*3[tex]\frac{2}{3}[/tex]). Finally subtract the area of the smaller figure from the area of the larger figure. ((7[tex]\frac{1}{4}[/tex]*6)-(5*3[tex]\frac{2}{3}[/tex])
Hope this helps!
The measure of an angle is 2°. What is the measure of its complementary angle?
Answer:
88°
Step-by-step explanation:
Complementary angles have a sum of 90°, just do 90 - 2 and there's your answer.
What is 2.5 × 15.625
Answer:
39.0625
Step-by-step explanation:
Answer:
39.0625
Step-by-step explanation:
convert 35 kg into lb, then round it to two decimal places
Answer:
77.16 lbs
:)))))))))))))
Gummi bears come in twelve flavors, and you have one of each flavor. Suppose you split your gummi bears among three people (Adam, Beth, Charlie) by randomly selecting a person to receive each gummi bear. Each person is equally likely to be chosen for each gummi bear.
(a) What is the probability that Adam gets exactly three gummi bears?
(b) If you know that each person recieved at least one gummi bear, what is the probability that Adam gets exactly three gummi bears?
(c) What is the expected number of gummi bears that Adam gets?
(d) If you know that each person recieved at least one gummi bear, what is the expected number of gummi bears that Adam gets?
(e) Is there more variance in the number of gummi bears that Adam gets with or without knowing that everyone gets at least one gummi bear? Explain your answer without calculations.
Step-by-step explanation:
From the given gummi bear, the chance that Adam is selected for any draw is 1/3 as well as the chance he is not selected at any draw is 2/3.
a). The probability of Adam getting exactly three gummi bears = P(Adam gets selected at 3 draws and not selected at the remaining 9 draws)
= [tex]$ (\frac{1}{3})^3 (\frac{2}{3})^9 = \frac{2^9}{3^{12}} $[/tex]
Now, the 3 draws where Adam gets selected can be any 3 out of 12 draws in [tex]$ {\overset{12}C}_3 $[/tex] = 220
Thus, probability of Adam getting three gummi bears = [tex]$ 220 \times \frac{2^9}{3^{12}} $[/tex]
= 0.21186
b). Probability that Adam will get the three gummi bears given each person will received at the most 1 gummi bear
= P(of the remaining 9 draws after assigning one gummi bear to each one, Adam gets selected at 2 draws and not selected at 7 draws) = [tex]$ {\overset{9}C}_2 (\frac{1}{3})^2 (\frac{2}{3})^7 $[/tex]
= 0.23411
c). Let X = Number of the gummi bears which Adam will get. Then, X = number of draws out of 12 draws Adam gets selected and X ~ B(12, 1/3). So, Adam will get gummi bears= mean of B(12, 1/3) = 12 x (1/3) = 4
d). Let Y = Number of the gummi bears that Adam will get, given each person will received at the most 1 gummi bear Then, Y = number of draws Adam gets selected in the remaining 9 draws and Y ~ B(9, 1/3). So, the expected number of bears that Adam gets given each person received at least 1 gummi bear
= 1 + mean of B(9, 1/3) = 4
e). When every one gets atleast one gummi bear, the new sample size will be 9 and so we can say that there is a reduction in variance.
A projectile is launched from ground level with an initial velocity of 89 ft/sec. It's height(ft), t seconds after launching is given by s=-16t^2 + v0t + s0. a) Fine the time(s) the projectile will reach a height of 100 feet. b) find the time it takes the projectile to hit the ground
Answer:
Step-by-step explanation:
Initial Velocity of projection = 89ft/sec
height(ft), t seconds after launching is given by s=-16t^2 + v0t + s0
A) time the projectile will reach a height of 100 ft
s=-16t^2 + v0t
v0 = 89ft/sec
s = 100ft
100 = - 16t² + 89t
-16t² + 89t - 100 = 0
Using the online quadratic equation solver;
t = 4s or t = 1.5625
x + y = -50
y = 4x
Answer (x,y)
Answer:
x = -10 ; y = -40
Step-by-step explanation:
Note:
y = 4x.
Plug in 4x for y in the first equation:
x + y = -50
x + (4x) = -50
Solve for x. First, simplify, then isolate the variable, x.
x + 4x = -50
5x = -50
Divide 5 from both sides of the equation.
(5x)/5 = (-50)/5
x = -50/5
x = -10
Plug in -10 for x in one of the equations:
y = 4x
y = 4(-10)
y = -40
x = -10 ; y = -40 is your answer.
~
What is the y-intercept of the linear equation x-3y=-6?
To find the y-intercept, let x = 0.
0 - 3y = -6
-3y = -6
y = 2
The y-intercept of the equation is the point (0, 2).
Answer:
2
Step-by-step explanation:
Solve the equation for y. First subtract x from both sides then divide by -3. The result is y=1/3x+2 with the number "2" representing the y-intercept.
You flip a fair coin until you see three tails in a row. What is the average number of headsthat you’ll see until gettingTTT?
Answer:
14
Step-by-step explanation:
From the given information:
suppose, Y is the number of times a coin is being flipped.
If the coin is flipped for the first time and we get H, then we have:
TTT = [tex]\dfrac{1}{2}(Y+1)[/tex]
Afterward, if we get H, then we waste two times plus the probability of this event [tex]\dfrac{1}{4}[/tex].
Therefore, we have : [tex]\dfrac{1}{4}(Y+2)[/tex]
Afterward, if we get H, then we waste three times plus the probability of this event [tex]\dfrac{1}{8}[/tex].
Therefore, we have : [tex]\dfrac{1}{8}(Y+3)[/tex]
If we got T at the third time, then;
T = [tex]\dfrac{1}{8}(3)[/tex]
Thus, average number of headsthat you’ll see until gettingTTT can be expressed as:
[tex]= \dfrac{1}{2}(Y+1)+ \dfrac{1}{4}(Y+2)+ \dfrac{1}{8}(Y+3)+ \dfrac{1}{8}(3)[/tex]
= 14
For what values of x will the result of 5 x be greater than 1?
Answer:
2&3
Step-by-step explanation:
2+3=5 or 3+2=5
You can continue on but you would need to use one and these are the two that are greater than 1 but make up 5.
The values of x for which the result of 5 x be greater than 1 would be 2 and 3.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.
If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
WE need to find the values of x for which the result of 5 x be greater than 1
WE can have that as;
2+3=5
or
3+2=5
Also we can continue on but we would need to use one and these are the two that are greater than 1 but make up 5.
Therefore, the values of x for which the result of 5 x be greater than 1 would be 2 and 3.
Learn more about the unitary method;
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what are the domain and range of the function represented by the set of order pairs? {(-12,-5),(-10,5),(10,-5),(12,5)}
Answer:
Domain: -12, -10, 10, 12
Range: -5, 5
Step-by-step explanation:
The Domain is the first coordinate which is x.
The Range is the second coordinate which is y.
Hope This Helps :)
PLZ HELP ASAP! ANSWER ALL FOR BRAINLIEST
Answer:
4.
-building
-artisting
-selling
-making good
-trading
-farming
5. "Being settled meant being tied to land and possessions; being nomadic meant having a mobile community with a mobile food supply. This allowed nomads to attack and plunder resources. They could gain access to agricultural products without having to farm or trade."
6. " They were also trained in riding and archery, as they sometimes had to defend their communities from outsiders. Women’s graves sometimes contained weapons, suggesting they had a role in military life as well."
Step-by-step explanation:
4. Ariel loves to go to baseball games. This season she has already attended 23 New York Ducks games. If
Ariel attended nine more weekend games than weekday games, how many weekday games did she attend.
Answer: She attended to 7 weekday games.
Step-by-step explanation:
Let's denote our variables as:
Wd = number of weekday games that she attended.
We = number of weekend games that she attended.
We know that in total, she attended to 23 games, so we have:
Wd + We = 23.
"If Ariel attended nine more weekend games than weekday games" can be written as:
We = Wd + 9.
Then we have the system of equations:
Wd + We = 23.
We = Wd + 9.
The first step would be to replace the second equation into the first one, as:
Wd + We = 23.
Wd + (Wd + 9) = 23
Now we can solve this for Wd.
2*Wd + 9 = 23
2*Wd = 23 - 9 = 14
Wd = 14/2 = 7
She attended to 7 weekday games.
And:
We + 7 = 23
We = 23 - 7 = 16
She attended at 16 weekend games.
The number of hours a group of 12 children in Mrs. Smith's kindergarten class read in a day.
Answer:
30 minuetes because their brain can take so many
Step-by-step explanation:
3y^2+y^- 5+ 4y+2y^3+8
Answer:
9y+3
Step-by-step explanation:
you add all the common numbers and you end up with:
9y^6+3+y^-5
then you put the y^-5 under the other numbers and make it positive so it becomes:
9y^6+3/y^5, then you subtract the exponents of y: 6-5, so that cancels out the y on the bottom making it:
9y+3
CR 3
Part A: Which sentence about integers is NOT ALWAYS TRUE (1 point)
A. positive -positive = positive
B. positive + positive = positive
C. negative + negative = negative
D. positive - negative = positive
Answer:
A. positive -positive = positive
Mr. Monasterio ran to The Burrito Barn at 10mph. He ate too many burritos and slowed to 6 mph on the way home. Find the time going to The Burrito Barn if the time coming back took 2 hours more than the time going. Don't forget to include units.
Answer:
it was 3 hours
Step-by-step explanation:
the thing said so
20% of ___= 100 Find the answer
Answer:
500
Step-by-step explanation:
20% of x = 100
Write 20% as a decimal - - > 0.2(x) = 100
Isolate and solve for x - - > x = 100/0.2
Therefore, x = 500
Answer:
500
Step-by-step explanation:
Let the no. be x
20% of x = [tex]\frac{20}{100} x[/tex]
And ,
[tex]\frac{20}{100} x = 100\\=> 20x = 10000\\ => x = \frac{10000}{20} = 500[/tex]
find the area of the region enclosed by the curves y=x and y=(x-2)^2
Answer:
Step-by-step explanation:
14.5845842 518.5874 funny
Answer:
Step-by-step explanation:
The heat in the house is set to keep the minimum and maximum temperatures (in degrees Fahrenheit) according to the equation |x – 72.5| = 4. What are the minimum and maximum temperatures in the house? 72.5°F and 76.5°F 68.5°F and 76.5°F 70.5°F and 74.5°F 72.5°F and 74.5°F
Answer:
The minimum and maximum temperatures in the house are 68.5°F and 76.5°.
To keep the house in heat, the minimum and maximum temperature is set to option 68.5°F and 76.5°F.
What is a word problem?A word problem is a verbal description of a problem situation. It consists of few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.
For the given situation,
The equation |x – 72.5| = 4.
The minimum and maximum temperatures can be found by substituting +4 and -4 in x.
Minimum temperature,
⇒ [tex]|+4-72.5|=|-68.5|[/tex]
⇒ [tex]68.5[/tex]
Maximum temperature,
⇒ [tex]|-4-72.5|=|-76.5|[/tex]
⇒ [tex]76.5[/tex]
Hence we can conclude that to keep the house in heat, the minimum and maximum temperature is set to option 68.5°F and 76.5°F.
Learn more about word problems here
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In a triangle, the measure of one angle is five more than the first and the third angle is one more than four times the first.
Answer:
A= 29
B=34
C=117
Step-by-step explanation:
A triangle has three sides and a triangle always adds up to be 180
A + B + C = 180
B = A + 5
C = 4A + 1
Plug in.
A + A + 5 + 4A + 1 = 180
6A + 6 = 180
6A = 174
A = 29
Now since you found A, find the other two by plugging A in!
C = 4(29) + 1
C = 117
Plug in the B equation
B = 29 + 5
B=34
Check your answers
29 + 117 + 34 = 180
write the sentence as a equations
the product of 248 and a is 391
Answer:
248*a=391
Step-by-step explanation:
Suppose we want to know whether students who study more (hours per week) have a higher grade point average. What kind of question is this? *
Answer:
See who studies and who doesnt then compare
Step-by-step explanation:
See who studies and who doesn't
take the assignment
ask for the score of the people who did and for who didn't
collect the data and organize it
what does x = in this equation? 7x-2+2x+14=33
Answer:
x = 7/3
Step-by-step explanation:
First, combine like terms:
7x + 2x - 2 + 14 = 33
(7x + 2x) + (14 - 2) = 33
9x + 12 = 33
Isolate the variable, note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.
First, subtract 12 from both sides:
9x + 12 (-12) = 33 (-12)
9x = 33 - 12
9x = 21
Next, divide 9 from both sides:
(9x)/9 = (21)/9
Simplify. Divide common factors:
x = (21/9)/(3/3) = 7/3
x = 7/3
Jamie will choose between two catering companies for an upcoming party. Company A charges a set up fee of $500 plus $25 for each guest. Company B charges a set up fee of $200 dollars plus 30per guest. How many guests are need where the price is equal
Answer:
60 guests
Step-by-step explanation:
We'll say that the number of guests is n.
We're looking for the n where both of these expressions are equal:
Company A (500 set-up fee, 25 for each guest)
500 + 25 * n
Company B (200 set-up fee, 30 for each guest)
200 + 30 * n
500 + 25n = 200 + 30n
Subtract 200 from both sides.
300 + 25n = 30n
Subtract 25n from both sides.
300 = 5n
We'll reverse the sides.
5n = 300
Divide both sides by 5.
n = 300/5 = 60
Thus, both companies are equal when n is 60, or when there are 60 guests.
helppppp i dont know pslßss
Answer:
width=3
Step-by-step explanation:
area=length X width
length=4 X width
=>area=4 X width X width
=>area=(2 X width)^2
=>36=(2 X width)^2
=>6=2 X width
=>width=3
the area of a rectangle is length times width
A = L times W
A = 26. so 36 = L times W
The length is 4 times the width
L = 4W
Thus, we can switch L to 4W in the area equation
36 = 4W ×W = 4W²
36 ÷ 4
=√9
=3cm
Consider the sequences of numbers.
5, 10, 15, 20, 25, 30, 35, 40, 45, . . .
10, 20, 30, 40, 50, . . .
Which statements are true regarding the patterns of the sequences? Check all that apply.
The first pattern involves repeated addition by 5.
The second pattern involves repeated addition by 10.
The first pattern involves multiplication by 2.
If a number is in the first pattern, then it is also in the second pattern.
The second pattern can be created by multiplying each number in the first pattern by 2.
Answer:
1,2,5
Step-by-step explanation:
The first pattern has a difference of 5 while the second is 10 and become by multiplying 2 in pattern 1 so options (A),(B), and (E) will be correct.
What is the pattern?A pattern is a specific arrangement of numbers or any term which will follow a specific code or hint.
The hint which is present in a given pattern hides in the pattern only.
For example 2,4,6,8 in this, the difference between any two-term is only 2 and this will be its specific pattern.
Another example is 2,4,16,256 in this pattern each term is the square of the previous term and this will become its specific pattern.
Given the two patterns.
In the first pattern, each term become by adding 5 so option (A) is correct.
In the second pattern, each term become by adding 10 so option (B) is correct.
The term in the second pattern becomes by multiplying 2 in pattern 1
For example 5× 2 = 10 and 10 × 2 = 20 so option (E) will be correct.
Hence "The first pattern has a difference of 5 while the second is 10 and become by multiplying 2 in pattern 1".
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Which of the following is NOT equivalent to 60/80?
A.)3/4 B.)75% C.)7.5 D.)0.750
Answer:
C
Step-by-step explanation:
Everything else is equivalent.
Hope this helps!
Answer:
C. 7.5
Step-by-step explanation:
60/80 = 3/4 = 75% = 0.75
Everything is equal except for 7.5 because 7.5 = 750%
Hope This Helps :)
Community Gym charges a $55 membership fee and a $40 monthly fee. Workout Gym
charges a $125 membership fee and a $30 monthly fee. How many months will it take for
the cost of both gyms to be the same? and what is the cost of the gyms during that number
of months?