The permutation equation [tex]^{n+1}P_3 - ^nP_3= 3 * ^nP_2[/tex] is true for all integers n ≥ 3
How to prove the equation?We have:
[tex]^{n+1}P_3 - ^nP_3= 3 * ^nP_2[/tex]
Apply the following permutation formula
[tex]^{n}P_r = \frac{n!}{(n- r)!}[/tex]
So, we have:
[tex]\frac{(n + 1)!}{(n + 1 - 3)!} - \frac{n!}{(n - 3)!} = 3 * \frac{n!}{(n - 2)!}[/tex]
Evaluate the difference
[tex]\frac{(n + 1)!}{(n - 2)!} - \frac{n!}{(n - 3)!} = 3 * \frac{n!}{(n - 2)!}[/tex]
Expand
[tex]\frac{(n + 1)!}{(n - 2)(n - 3)!} - \frac{n!}{(n - 3)!} = 3 * \frac{n!}{(n - 2)(n - 3)!}[/tex]
Multiply through by (n - 3)!
[tex]\frac{(n + 1)!}{(n - 2)} - n! = 3 * \frac{n!}{(n - 2)}[/tex]
Expand
[tex]\frac{(n + 1) * n!}{(n - 2)} - n! = 3 * \frac{n!}{(n - 2)}[/tex]
Divide through by n!
[tex]\frac{(n + 1)}{(n - 2)} - 1 = \frac{3}{(n - 2)}[/tex]
Take the LCM
[tex]\frac{n + 1 - n + 2}{(n - 2)} = \frac{3}{(n - 2)}[/tex]
Evaluate the like terms
[tex]\frac{3}{(n - 2)} = \frac{3}{(n - 2)}[/tex]
Both sides of the equations are the same.
Hence, the permutation equation [tex]^{n+1}P_3 - ^nP_3= 3 * ^nP_2[/tex] is true for all integers n ≥ 3
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Select Translation, Reflection, or Rotation to identify the single transformation that transforms ∆PQR to ∆P'Q'R'.
Translation Reflection Rotation
(The pictures of the graphs are in the file attached, please look at them to know which ones are for which option.
Which table represents a proportional relationship? A 2-column table with 3 rows. Column 1 is labeled x with entries 4, 8, 18. Column 2 is labeled y with entries 6, 12, 26. A 2-column table with 3 rows. Column 1 is labeled x with entries 5, 20, 60. Column 2 is labeled y with entries 7, 28, 84. A 2-column table with 3 rows. Column 1 is labeled x with entries 12, 18, 26. Column 2 is labeled y with entries 5, 7, 9. A 2-column table with 3 rows. Column 1 is labeled x with entries 3, 9, 15. Column 2 is labeled y with entries 2, 8, 14.
Answer:
X Y
5 7
20 28
60 84
This table shows the proportional relationship.
Step-by-step explanation:
The equivalent ratios of two variables known as proportional relationship.
Table 1: X Y Table 3: X Y
4 6 12 5
8 12 18 7
18 26 26 9
Ratio = 4:6 = 8:12 ≠ 18:26 Ratio = 12:5 ≠ 18:7 ≠ 26:9
Table 2: X Y Table 4: X Y
5 7 3 2
20 28 9 8
60 84 15 14
Ratio = 5:7 =20:28=60:84 Ratio = 3:2 ≠ 9:8 ≠ 15:14
Here we see table 2 shows the proportional relationship.
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What function does this graph represent?
What is the value of f(1)?
0
1
2
4
Answer:
Solution*-1
Step-by-step explanation:
because 1 is the value of (1)
3x + 2(x − 1) − 4? pleeeessssssssssssssssssssssssssss
Answer:
5x - 6Step-by-step explanation:
3x + 2(x − 1) − 4 =
3x + 2x - 2 - 4 =
5x - 6
Answer:
[tex]\boxed{\bf 5x - 6}[/tex]Step-by-step explanation:
[tex]\bf 3x + 2(x - 1) - 4?[/tex]
Distribute.
[tex]\bf 3x + 2x - 2 - 4[/tex]Subtract the numbers.
[tex]\bf 3x + 2x - 2 - 4[/tex][tex]\bf 3x + 2x - 6[/tex]Combine like terms.
[tex]\bf 5x - 6[/tex]_____________________Help please.. z zzz z Z z z z z z z z z z z z z zz
Answer:
B
Step-by-step explanation:
look at the graph, it is really easy with such basic numbers involved :
the red dotted line is
y = 2
since the line is dotted (= the points on the line are not included in the solution), the colored area is below the line is
y < 2
the blue line is clearly just a 45° basic line through the origin (0,0), and that means for every point in that line x is equal to y.
y = x
and the line is solid, so the points are included in the solution. the shaded area is above the line, so, we need to use the ">" relation :
y >= x
the required solution is the combination of both, and therefore, B is correct.
Which expression is equivalent to 4 √10 ?
(૧/x
0 x ²
022
○ +³² (+ √/x)
0 +5
ANSWER FAST PLEASE!!!
Which of the following systems of inequalities would produce the region
indicated on the graph below?
+11
10
Jadosak
-1
2 3
15
6
7
8 9
A. v2x+1; y210; x ≥ 0
B. v2x+1; y<10; x ≥ 0
O c. v
OD. V > x+1; y≤ 10; x ≥ 0
Answer: D
Step-by-step explanation:
We know that the line [tex]y=10[/tex] is shaded below, so it represents [tex]y \leq 10[/tex].
Similarly, the line [tex]y=x+1[/tex] is shaded above, so it represents [tex]y > x+1[/tex].
Complete the steps in solving the quadratic function 7x – 9 = 7x2 – 49x by completing the square.
answer below:
–9 = 7x2 –
⇒ 56 x
–9 +
⇒ 112 = 7(x2 – 8x +
⇒ 16 )
56, 112, 16 in that order
Answer:
it's actually 28+/-√721/7
Step-by-step explanation:
46. Geometry The volume of a cube is a function of the length of a side of the cube. Write a function for the volume of a cube. Find the volume of a cube with a side 13.5 cm long.
Answer:
1/3×22/7×r^2h
Step-by-step explanation:
the volume =1/3 ×22/7×r^2×13.5
What are the vertex, focus, and
directrix of the parabola with
equation y-6=2(x-4)²?
Answer:
vertex (2,6)focus(49/8,2)directrix y=47/8can someone help me here pls thanks!
Using proportions, it is found that the measure of the inscribed angle B is of 30º.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
In a circle, the inscribed angle is half the measure of the outside angle. Hence, in this problem, the measure of angle B is of 50% of 60º, hence:
m<B = 0.5 x 60º = 30º.
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Answer:
Using proportions, it is found that the measure of the inscribed angle B is of 30º.
Step-by-step explanation:
Is 200 a term in the sequence 4, 10, 16, 22,
Answer:
Step-by-step explanation:
If T: (x, y) → (x + 6, y + 4), then T-1: (x,y) → _____.
( x/6, y/4 )
( x - 4, y - 6)
(-6 x, -4 y)
( x - 6, y - 4)
T is a linear transformation from R²→R² with basis {(1,0),(0,1)}
T: (x,y)→(x+6,y+4)
A function from one vector space to another that preserves the underlying (linear) structure of each vector space is called a linear transformation.
Then the vector (1,0) goes to (1+6,4)=(7,4)=7(1,0)+4(0,1)
and the vector (0,1) goes to (6,1+4)=(6,5)=6(1,0)+5(0,1)
So, the matrix of the transformation is
[tex]\left[\begin{array}{ccc}7&6\\4&5\end{array}\right][/tex]
The inverse of the matrix is
[tex]\left[\begin{array}{ccc}\frac{5}{11}&\frac{-6}{11}\\ \\\frac{-4}{11}&\frac{7}{11}\end{array}\right][/tex]
So, the Inverse Transformation is given by
[tex]T^{-1}(x,y)=\left[\begin{array}{ccc}\frac{5}{11}&\frac{-6}{11}\\ \\\frac{-4}{11}&\frac{7}{11}\end{array}\right]\left[\begin{array}{ccc}x\\y\end{array}\right] =(\frac{5x-6y}{11}, \frac{-4x+7y}{11})[/tex]
So, no option is correct. And the answer is
[tex]T^{-1}(x,y)=(\frac{5x-6y}{11}, \frac{-4x+7y}{11})[/tex]
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are these right tell me yes or no
Answer:
Question 4: letter c
Question 3: letter c
Step-by-step explanation:
What is the effect on the graph of the function f(x) = x2 when f(x) is changed to f( 1 3 x) + 6? A) stretched horizontally and shifted up B) stretched vertically and shifted right C) compressed vertically and shifted left Eliminate D) compressed horizontally and shifted down
Answer:
A
Step-by-step explanation:
f(x) = x²
f(x) becomes f(13x) + 6
the "+ 6" at the end tells us already everything, because there is only one aber option listed that contains a shift up. and that is exactly what that "+ 6" did. it adds 6 to every possible y value of the function. so, the whole function moves up by 6 units.
and yes, the 13x of f(13x) stretches the function horizontally (along the x axis).
simply said, whenever the x value for the original function increased by 1, it increases for the new function by 13.
please help fast grade 5 math extra points to those who explain and giving brainliest
Answer:
[tex]\frac{45}{7}[/tex]
Step-by-step explanation:
9 is the same as [tex]\frac{9}{1}[/tex]
∴ 9 ˣ [tex]\frac{5}{7}[/tex]
= [tex]\frac{9}{1}[/tex] ˣ [tex]\frac{5}{7}[/tex]
To multiply two fractions, just multiply their numerators together and their denominators together.
9 x 5 = 45
1 x 7 = 7
∴ [tex]\frac{9}{1}[/tex] ˣ [tex]\frac{5}{7}[/tex]
= [tex]\frac{45}{7}[/tex]
what is equivalent to 7/5y
Answer:
1.4y
Step-by-step explanation:
1. Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Choosing 5 people (without replacement) from a group of 58 people, of which 15 are women, keeping track of the number of men chosen.
A) Not binomial: there are too many trials.
B) Procedure results in a binomial distribution.
C) Not binomial: there are more than two outcomes for each trial.
D) Not binomial: the trials are not independent.
The 15 are women, keeping track of the number of men chosen is Not binomial the trials are not independent.
We have given that,
The given procedure results in a binomial distribution. If not, state the reason why. Choosing 5 people (without replacement) from a group of 58 people, of which 15 are women, keeping track of the number of men chosen.
What is the Binomial distribution?In samples with replacement, the probability has to be the same for each trial, that is, they are independent.
Hypergeometric distribution
Samples without replacement, as the probabilities depend on the previous outcomes, that is, they are not independent.
In this question
Marbles are chosen without replacement, which means that for each trial the probabilities of choosing a red marble are different, which means that the trials are not independent. So the answer is:
Not binomial the trials are not independent.
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Need help with this question (pic included)
Answer:
answer= 3
hope this helps if not I'll try to do it again
\How could Brent use a rectangle to model the factors of x2 – 7x + 6?
He could draw a diagram of a rectangle with dimensions x – 3 and x – 4 and then show the area is equivalent to the sum of x2, –3x, –4x, and half of 12.
He could draw a diagram of a rectangle with dimensions x + 7 and x – 1 and then show the area is equivalent to the sum of x2, 7x, –x, and 6.
He could draw a diagram of a rectangle with dimensions x – 1 and x – 6 and then show the area is equivalent to the sum of x2, –x, –6x, and 6.
He could draw a diagram of a rectangle with dimensions x – 4 and x + 3 and then show the area is equivalent to the sum of x2, –4x, 3x, and half of –12.
Option third "He could draw a diagram of a rectangle with dimensions x – 1 and x – 6 and then show the area is equivalent to the sum of x², –x, –6x, and 6' is correct.
What is the area of the rectangle?It is defined as the area occupied by the rectangle in two-dimensional planner geometry.
The area of a rectangle can be calculated using the following formula:
Rectangle area = length x width
We have a quadratic expression to model the area of the rectangle:
= x² – 7x + 6
[tex]\rm =\left(x^2-x\right)+\left(-6x+6\right)[/tex]
[tex]=\rm x\left(x-1\right)-6\left(x-1\right)[/tex]
= (x - 1)(x - 6)
Thus, option third "He could draw a diagram of a rectangle with dimensions x – 1 and x – 6 and then show the area is equivalent to the sum of x², –x, –6x, and 6' is correct.
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Select the correct answer.
In the given figure, m AD = 79° and m
Answer:
2 or 3
Step-by-step explanation:
write it down dummy
Rewrite in simplest terms: (4x+5)+(−5x−1) pls help
Answer:
-x + 4
Step-by-step explanation:
(4x+5)+(−5x−1)
4x+5−5x−1 [Remove the parentheses]
-x + 4 [Combine Like Terms]
Answer:
-x+4
Step-by-step explanation:
Given:
(4x+5)+(−5x−1)
Solution:
To re-write this in it's simplest terms:
4x+5-5x−1Combine like terms:
4x-5x+5-1-x+4Hence the simplest form is -x+4.
Done.
Use logarithmic properties and solve for x
Answer:
[tex]x = 45 \sqrt{5} [/tex]
Step-by-step explanation:
[tex] {3}^{2 log_{3}(x) + log_{3}(5) } = {3}^{ 4log_{3}(15) } [/tex]
[tex] {3}^{ log_{3}( {x}^{2} ) } {3}^{ log_{3}(5) } = {3}^{ log_{3}( {15}^{4} ) } [/tex]
[tex]5 {x}^{2} = {15}^{4} [/tex]
[tex] {x}^{2} = \frac{ {15}^{4} }{5} [/tex]
[tex]x = \frac{ {15}^{2} }{ \sqrt{5} } ( \frac{ \sqrt{5} }{ \sqrt{5} } )[/tex]
[tex]x = \frac{ {15}^{2} \sqrt{5} }{5} = 45 \sqrt{5} [/tex]
How does the graph of g(x)=
X-5
+2 compare to the graph of the parent function f(x) = ?
g(x) is shifted 5 units left and 2 units up from f(x).
g(x) is shifted 5 units right and 2 units up from f(x).
Og(x) is shifted 5 units left and 2 units down from f(x).
Og(x) is shifted 5 units right and 2 units down from f(x).
Answer:
B. g(x) is shifted 5 units right and 2 units up from f(x).
g(x) = [tex]\frac{1}{x-5}[/tex] + 2 is compared to the parent function, f(x) = [tex]\frac{1}{x}[/tex] by g(x) is shifted 5 units right and 2 units up from f(x).
What is Translation of Functions?Translation of functions is defined as the when each point in the original graph is moved by a fixed units in the same direction.
There are horizontal translation and vertical translation of functions.
Given parent function is,
f(x) = [tex]\frac{1}{x}[/tex]
And the translated function is,
g(x) = [tex]\frac{1}{x-5}[/tex] + 2
When f(x) becomes g(x), the first change occurred is [tex]\frac{1}{x}[/tex] changes to [tex]\frac{1}{x-5}[/tex].
That is f(x) changes to f(x-5).
This translation occurs when the function undergoes horizontal translation towards right.
Then [tex]\frac{1}{x-5}[/tex] is translated to [tex]\frac{1}{x-5}[/tex] + 2.
That is, f(x-5) is changed to f(x-5) + 2.
This occurs for the vertical translation towards up.
Hence the translation occurred is g(x) is shifted 5 units right and 2 units up from f(x).
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(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 .
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.
AandBhave coordinates (-1,1.5) and (3,2).
Tangents to the curve have been drawn at A and B.
Calculate the gradient of the curve at A and at B.
What is the slope of a line that is parallel to the line y 3/4x+2
Answer:
-4/3
Step-by-step explanation:
mtan × mline = -1
mtan× 3/4 = -1
mtan =-4/3
whats the y-inter of the line? interpet the y-inter in the context of the problem
Answer:
How do you interpret the y-intercept in the context of the problem?
The y-intercept of a line is the value of y where the line crosses the y-axis. In other words, it is the value of y when the value of x is equal to 0. Sometimes this has true meaning for the model that the line provides, but other times it is meaningless.
Step-by-step explanation:
A shop increases all their prices by 6%
A picture frame increases by £294
Find the new price of the picture frame.
Answer:
0riginal price = 4900
Step-by-step explanation:
294 = 6%
294/6 = 49 49 = 1%
49 x 100 = 4900
0riginal price = 4900
