Answer:
Proof below.
Step-by-step explanation:
Let f(n)=10+ n^2 - (n - 2)^2.
We want to prove f is even for integer n > or = 1.
Let's show f is even for the base case, n=1.
f(1)=10+1^2-(1-2)^2
f(1)=10+1-(-1)^2
f(1)=11-(1)
f(1)=10
f(1) is even.
Let's assume the for some integer k > or = 1 , we have f(k) is even. This means there is some integer m so that f(k)=2m.
Note: f(k)=10+ k^2 - (k - 2)^2.
We want to show f(k+1) is even.
f(k+1)=10+ (k+1)^2 - ([k+1] - 2)^2
f(k+1)=10+ (k+1)^2 - (k-1)^2
f(k+1)=10+ ((k+1)^2 - (k-1)^2)
That difference is a difference of squares. Let's try factoring it.
f(k+1)=10+ (k+1-[k-1])(k+1+[k-1])
f(k+1)=10+ (2)(2k)
Multiply:
f(k+1)=10+4k
Turns out we didn't even need to use our induction hypothesis since 10+4k is 2(5+2k) which makes f(k+1) even given the factor of 2 present.
This proves for all integers n>=1 that f(n) is even.
We could have this without induction then. Let's do that here below this line.
-------------------
10+ n^2 - (n - 2)^2
10+[n^2 - (n - 2)^2]
Factor difference of squares:
10+(n-(n-2))(n+(n-2))
Simplify:
10+(2)(2n-2)
Factor:
2[5+(2n-2)]
Because of the factor of 2, the given expression is always even for integer n.
what is the formula for calculating midpoint
The intersection of a line segment's two endpoints is known as the midpoint. From the midpoint to one endpoint and the midpoint to the other endpoint, you will be given two equal distances.
The formula for calculating the midpoint is given below:
[tex] \boxed{ \bold{ \: \: M = \bigg(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\bigg) \: \: }}[/tex]For example, let us try solving this sample problem below using the midpoint formula:
Sample problem: A new transmission tower will be put up midway between two existing towers. On a map drawn on a coordinate plane, the coordinates of first existing tower are (-5, -3) and the coordinates of the second existing tower are (9, 13). What are the coordinates of the point where the new tower will be placed?
From inspection on the given problem:
[tex]\sf (x_1, y_1) = (-5, -3) [/tex][tex]\sf (x_2, y_2) = (9, 13) [/tex]Substitute the given values into the midpoint formula and solve for M:
[tex]\sf M = \bigg(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\bigg)[/tex][tex]\sf M = \bigg(\dfrac{-5 + 9}{2}, \dfrac{-3 + 13}{2}\bigg)[/tex][tex]\sf M = \bigg(\dfrac{4}{2}, \dfrac{10}{2}\bigg)[/tex][tex]\sf M = (2, 5)[/tex]Therefore, the coordinates of the point are (2, 5).
[tex]\\[/tex]
Now, wasn't that easy? Try this other sample problem:
The endpoints of a segment are (-5, 2) and (9, 12), respectively. What are the coordinates of its midpoint?From the sample problem above, you may solve it on your own to enhance your skill on calculating the midpoint.
Cheers!
In a standard deck of 52 playing cards, 13 cards are clubs, and 3 of the clubs are "face" cards (K, Q, J). What is the probability of drawing one card that is:
A) A club or a face card? Is this event a union or an intersection?
B) A club and a face card? Is this event a union or an intersection?
C) Not a club and not a face card?
A) This event is a union. If a card is of the suit of clubs or a face card or both, then it belongs to the set of "cards with clubs or a face".
For the probability in question, we use the inclusion/exclusion principle:
P(club or face) = P(club) + P(face) - P(club and face)
That is, we add the probabilities of drawing a club and of drawing a face card, and subtract the probability of both conditions being met because, in a certain sense, we've double-counted the event of the card being both of clubs and a face. The sets "club" and "face" are not disjoint; there is overlap between them.
Then we have
P(club) = 13/52
P(face) = 12/52
P(club and face) = 3/52
so that
P(club or face) = 13/52 + 12/52 - 3/52 = 22/52 = 11/26
B) This event is an intersection. If a card is of the suit of clubs and a face card, then it belongs to both the set of clubs and the set of face cards.
We already found the probability of the intersection above:
P(club and face) = 3/52
C) If a given card is not a club and not a face card, this is the same as saying the card is neither a club nor a face card. In other words,
P(not club and not face) = P(not(club or face))
and this probability is complementary to the probability that a card is either a club or face, which is to say
P(not(club or face)) = 1 - P(club or face)
Using the result from part (A), we have
P(not club and not face) = 1 - 11/26 = (26 - 11)/26 = 15/26
euler formula find the number of edges of a polyhedron of 6 faces and 4 verticies
Euler's polyhedron formula, V − E + F = 2
[tex] = 4 - E + 6 = 2 \\ = E - 10 = 2 \\ = E = 2 + 10 \\ = E = 12[/tex]A line with a slope of -6 passes through the point (-8, -3). What is its equation in point-
slope form?
Use the specified point in your equation. Write your answer using integers, proper fractions,
and improper fractions. Simplify all fractions.
Answer:
y+3=-6(x+8)
Step-by-step explanation:
Because we already know a point and the slope, we can put our equation in the form y-y1=m(x-x1);
y+3=-6(x+8)
❤️WILL REPORT BOTS❤️
Describe function transformations KT8
What kind of transformation converts the graph of
f(x) = 3x - 9
into the graph
of g(x) =-3x - 9?
Answer:
The answer is reflection on the y-axis
Step-by-step explanation:
Reflection on the x-axis will change the y-intercept
Horizontal shrink will change the slope but not to the point of making the 3 negative
Horizontal stretch is the same concept
Thus Reflection on the y-axis on the y-axisis the answer
Hope this helps!
Determine the following probabilities. Enter your final answers as reduced fractions. Two coins are flipped. Find the probability that the first coin land on heads and the second coin lands on heads. Two cards are drawn from a deck of 52 cards. The first card is replaced before drawing the second card. Find the probability that the first card is black and the second card is a 8. A single digit between 0 and 9 is randomly chosen, and a single letter from A to Z is randomly chosen. Find the probability that the number is 4 and the letter is a consonant. Two dice are rolled. Find the probability that first die lands on an even number and the second die is less than 2.
The probability of an event occurring is given by the ratio of the number of
possible outcome to the number of required outcome.
First question: The probability that the first coin lands on heads and the second coin lands on tails is 0.25.Second question: The probability of drawing a black card and then a 8 is [tex]\underline{\dfrac{1}{13}}[/tex].Third question: The probability that the number chosen is 4 and the letter chosen is a consonant, is [tex]\underline{\dfrac{21}{234}}[/tex].Fourth question: The probability that the first die lands on an even number and the second die is less than 2, is [tex]\underline{\dfrac{1}{12}}[/tex].Reasons:
First question:
The number of faces in a coin = 2; A head or a tail
The probability that the first coin lands on heads, P(H) = [tex]\dfrac{1}{2}[/tex]
The probability that the second coin lands on tails, P(T) = [tex]\dfrac{1}{2}[/tex]
The probability that the first coin lands on heads and the second coin lands
on tails = P(H ∩ T)
Which gives;
[tex]P(H \cap T) = \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}[/tex]
The probability that the first coin lands on heads and the second coin lands on tails = [tex]\dfrac{1}{4}[/tex] = 0.25
Second question:
The number of black cards in a pack of 52 = 26 cards
The number of cards that are a 8 in a pack of 52 cards = 8 cards
[tex]\mathrm{The \ probability \ of \ drawing \ a \ black \ card}, \ P(B) = \dfrac{26}{52} = \dfrac{1}{2}[/tex]
[tex]\mathrm{The \ probability \ of \ drawing \ a \ 8,} \ P(8) = \dfrac{8}{52} = \dfrac{2}{13}[/tex]
The probability of drawing a black card and then an 8, P(B∩8), is given as follows;
[tex]P(B \cap 8) = \dfrac{1}{2} \times \dfrac{2}{13} = \dfrac{1}{13}[/tex]
The probability of drawing a black card and then a 8 is P(B∩8) = [tex]\underline{\dfrac{1}{13}}[/tex]
Third question:
The probability that a number chosen between 0 and 9 is 4, P(4) = [tex]\dfrac{1}{9}[/tex]
The number of consonant in the alphabet = 21
The probability that a letter chosen from A to Z is a consonant, P(C) = [tex]\dfrac{21}{26}[/tex]
The probability that the number chosen is 4 and the letter chosen is a consonant, P(4 ∩ C) = [tex]\dfrac{1}{9} \times \dfrac{21}{26} = \underline{ \dfrac{21}{234}}[/tex]
Fourth question:
The number of even numbers on a die = 3; (2, 4, 6)
The number of numbers less than 2 on a die = 1
The probability that the first die lands on an even number, P(E) = [tex]\dfrac{3}{6}[/tex]
The probability that the second die is less than 2. P(<2) = [tex]\dfrac{1}{6}[/tex]
Therefore;
The probability that the first die lands on an even number and the second die is less than 2, P(E ∩ <2) = [tex]\dfrac{3}{6} \times \dfrac{1}{6} = \dfrac{3}{36} = \underline{\dfrac{1}{12}}[/tex]
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What is the average rate of change of the function f(x)=120(0. 1)x from x = 0 to x = 2? Enter your answer, as a decimal, in the box. Do not round your answer.
The average rate of change of the function is -59.4
The formula for calculating the rate of change of the function is expressed as:
[tex]f'(x)=\frac{f(b)-f(a)}{b-a}[/tex]
Given the function
[tex]f(x)=120(0.1)^x\\f(0) = 120(0.1)^0\\f(0)=120\\[/tex]
Get the value of f(2)
[tex]f(2)=120(0.1)^2\\f(2) = 120(0.1)^2\\f(2)=1.2\\[/tex]
Substitute into the formula to have:
[tex]f'(x)=\frac{1.2-120}{2-0}\\f'(x)=\frac{118.8}{2} \\f'(x)= -59.4[/tex]
Hence the average rate of change of the function is -59.4
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Suppose the area of a triangle is 36 square inches. The height of the triangle is 12 inches. What is the length of the base of the triangle? PLSSSSSS HELP MEEEEEEEEEEEEEEEEEEEEE
Answer: The base is going to be 6 inches.
Step-by-step explanation:
So to find the area of a triangle we need to know the formula which is A= 1/2bh (A= one half times base times height) So, all we need to do is multiply the area by 2
32*2=72 Then to find the missing length, we divided the area by 12 which is 6.
Find the y-intercept and then write the linear equation for the line passing
through (-12, 5) with m = -2/3,
Answer:
Y-intercept: -3
Equation: y = -2/3x -3
Step-by-step explanation:
Slope intercept equation: y = mx + b, m is the slope and b is the y-intercept
Substitute m with -2/3, your equation should now be y = -2/3x + b
To find the y-intercept, plug in the point (-12,5) and solve for b.
1. 5 = -2/3(-12) + b
2. 5 = 8 + b
3. b = 8 - 5
4. b = 3
A countertop is 13 feet long and 2 feet wide.
What is the area of the countertop in square meters? Use the conversion 1 foot = 0.305 meter. Round your final answer to 2 decimal places.
Answer:
A = 2.42 sq. meters
Step-by-step explanation:
13 feet = 13 × 0.305 = 3.965 meter
2 feet = 2 × 0.305 = 0.61 meter
A = lw
A = (3.965) × (0.61)
A = 2.41865 sq. meters
A = 2.42 sq. meters
9i⋅(−4−7i) Your answer should be a complex number in the form a+bi where a and b are real numbers.
Answer:
see below
Step-by-step explanation:
here we are using the distributive property and noting that i^2=-1 so the answer is -36i-(63i^2) or -36i+63
What is countercurrent multiplication?.
Answer:
What is countercurrent multiplication? ... Countercurrent multiplication in the kidneys is the process of using energy to generate an osmotic gradient that enables you to reabsorb water from the tubular fluid and produce concentrated urine.
Find the value of z
Answer:
z = 9 I think
Step-by-step explanation:
Because I did the quiz
Find the value of z.
7
Х
3
Z
Z =✓[?
Applying the leg rule, the value of z in the right triangle given is: √30 units.
What is the Leg Rule?The leg rule states that, hypotenuse/leg = leg/part.
Given the following:
Leg = z unitsPart = 3 unitsHypotenuse = 3 + 7 = 10 unitsPlug in the values into the leg rule:
10/z = z/3
z² = (10)(3)
z² = 30
z = √30 units
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2. Find the value of x. Give your answer in
simplest radical form.
20
16
Answer:
12
Step-by-step explanation:
The Pythagorean theorem shows that 400 or 20² - 256 or 16² = 144 or 12² which means that the other side is 12
x=12
For his car, Benny paid $70.96 on speakers and $83.52 on new tires with
wo $100 bills. How much did Benny get in change?
Answer:
If he pays with 2 Hundred Dollar Bills he will get $45.52 in change.
Explanation:
$70.96 + $83.52 = $154.48
2 Hundred Dollar Bills = $200.00
$200.00 - $154.48
What is the exponential form of the logarithmic equation?
4=log0.7(0.2401)
The exponential form of the logarithmic equation 4 = [tex]log_{0.7} (0.2401)[/tex] is
0.7⁴ = 0.2401
4 = [tex]log_{0.7} (0.2401)[/tex]
The exponential form of the logarithmic equation can be represented below:
Using logarithm rules,
logₐ b = x
Can be represented in exponential form as follows:
aˣ = b
Therefore,
4 = [tex]log_{0.7} (0.2401)[/tex]
using the logarithm rule described above can be represented in exponential form as follows:
0.7⁴ = 0.2401
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Answer:
0.7⁴ = 0.2401
Step-by-step explanation:
The temperature of a substance in an experiment changes by 8.7F from 10 am to 1p.m. At 5p.m., the temperature is 33.5°F. It
is 2/3 of what the temperature was at 1p.m.
What was the temperature at 10 a.m. of the substance?
Enter your answer as a decimal in the box
Answer:
138
Step-by-step explanation:
What's ( Round to the indicated place value
1. 6085 to the nearest hundred
2. 51,672 to the nearest ten thousand
3. 321.6396 to the nearest thousand)
Answer:
I'm not going to give the answer yet to see if you can figure it out.
Step-by-step explanation:
1. Find the place that your are rounding to. Ex: 6085
2. Then, look "nextdoor"; the number to the right of the number underlined. Ex: 6085
3. If this number is 1-4, then keep it the same. If it is 5-9, then add one to the digit to the left. Ex: 6085 becomes 6185
4. All of the digits to the right stay the same. Ones to the left become zero. Ex: 6185 becomes 6100.
I hope this helps!
The perimeter of the table shown is 16 feet. Write an equation in the form px+q=r to solve for x.
p = 2
q = 6
r = 16
x = 5 feet
Step-by-step explanation:
The perimeter of the rectangle is
In your case,
Width = 3 feet
Length = x feet
So, the perimeter is
Since the perimeter is 16 feet, we have
Hence,
p = 2
q = 6
r = 16
Solve this equation for x:
Graph the equation.
4) Monica swam 4 laps in 15 minutes. At this rate, how many laps would she swim in 45 minutes?
O 8 laps
O 12 laps
O 15 laps
o 21 laps
Answer:
12 laps
Step-by-step explanation:
15÷4=3.75
45÷12=3.75
not sure whether it's true or not but at least I got the answer-
What is the vertex and p-value of this parabola?
Answer:
vertex: (-7, 5)p = 2Step-by-step explanation:
The vertex is halfway between the focus and the directrix:
((-7, 7) +(-7, 3))/2 = (-14, 10)/2 = (-7, 5) . . . . vertex coordinates
The value of p is the distance from the focus to the vertex:
7 -5 = 2 = p
In vertex form, the equation of the parabola is ...
y = 1/(4p)(x -h)^2 +k . . . . . . vertex (h, k)
y = 1/8(x +7)^2 +5
The LEADING COEFFICIENT OR THE LEADING TERM of 3x - 5x^2y^3 + 6xy + 7 is
A)3x
B) - 5x^2y^3
C)5x^2y^3
D) 6xy
NONSENSE=REPORT
The leading term is the term with the highest degree (highest exponent number). Here, the leading term is =》B) - 5x²y³
³ is the highest degree in the given expression.
_____
Hope it helps ⚜
please help what is #4?
To Prove:-
JK=IJ
JK= ½IK = IJ= ½IK
JK= IJ(Equal Distance)
GK//HI (Parallel)
Therefore, JK congruent IJ
Hope This Helps
PLS HELP ILL GIVE BRAINLIEST
Kim's age is twice that of her sister. When you add Kim's age to her sister's age, you get 30. How old is each sister?
(a) Write an equation that represents the situation. Explain any variable used.
(b) Solve the equation from Part (a). Show your work. State your solution as a complete sentence.
Answer:
Kim's age is twice of that of her sister. We'll let k2 = Kim and k = Kim's sister.
So, k2. When you add Kim's age to her sister age you get 36.
Because Kim is twice the age of her sister k2. And because it said 'add' then we put a + sign. k2 + s = 36. Now let's solve it.
k2 + k = 36 < Given
3k = 36 < Combine like terms
k = 12 < Divide both sides by 3
We know Kim's sister is 12 y/o but what about Kim herself? Earlier we assigned Kim to k2. Substitute 12 for k and you get 12(2) = 24
Kim's sister is 12 y/o and Kim is 24 y/o
Step-by-step explanation:
Mark as brainliest please
Kim's age = x
Age of her sister = y
Acoording to conditions, [tex]x=2y[/tex]
The equation is,
[tex]x+y = 30 \\\\\implies 2y + y = 30\\\\\implies 3y = 30 \\\\\implies y = 10\\\\x = 2y = 2(10) = 20\\\\\text{Hence, Kim is 20 years old and her sister is 10 years old.}[/tex]
I don't understand this question can someone help me with this please, please explain!! and thank you for the help!! If u dont know try your best. Please do it quick :D
Answer:
No, it is not a prime factorization, this is the real prime factorization
Step-by-step explanation:
2⁵x 3 x 5⁴ x 11
The first one was not a prime factorization because 8 is not a prime number.
i need help with this
Answer:
88
Step-by-step explanation:
The triangles are congruent. Also, a triangle has a sum of 180 degrees for its interior angles.
Angles D and A are both unknown.
Angles O and P are both 55.
Angles G and W are both 37.
37+55 = 92
That means there would be 88 degrees left for the unknown angle to get to the total of 180 degrees.
92 + 88 = 180
Jan is looking for similar figures. Which choice is similar to the figure shown?
Bottom left as it's measurements and shape are the closest to the one Jan has
A 4-mile cab ride cost $5.50, and a 9 mile cab ride costs $8.00. Identify two points from this information. Let the total cost of the cab ride represent (y) and the number of miles traveled (x)
Two points which can be identified from this information are; (4, 5.50) and (9, 8.00).
According to the question;
total cost of the cab ride represent (y) and the number of miles traveled (x).In essence, the information provided can be used to provide two points as follows; (4, 5.50) and (9, 8.00).
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