Answer:
Step-by-step
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Mark brainliest if this answer is correct please
4 is the mode because 4 happens three times
Hey there! :)
Answer:
![n_{4} = 8\\ n_{5} = 16](https://tex.z-dn.net/?f=n_%7B4%7D%20%3D%208%5C%5C%20n_%7B5%7D%20%3D%2016)
Formula: ![f(n) = 2^{n-1}](https://tex.z-dn.net/?f=f%28n%29%20%3D%202%5E%7Bn-1%7D)
Step-by-step explanation:
Derive a rule from the numbers in the sequence:
![n_{1} = 1\\n_{2} = 2\\n_{3} = 4](https://tex.z-dn.net/?f=n_%7B1%7D%20%20%3D%201%5C%5Cn_%7B2%7D%20%3D%202%5C%5Cn_%7B3%7D%20%3D%204)
We can see that each number is double of the previous. We can write an explicit function describing this sequence:
where 'n' is the term number.
Substitute in to solve for the 4th and 5th terms:
![f(4) = 2^{4-1} = 2^{3} =8](https://tex.z-dn.net/?f=f%284%29%20%3D%202%5E%7B4-1%7D%20%20%3D%202%5E%7B3%7D%20%3D8)
![f(5) = 2^{5-1} = 2^{4} =16](https://tex.z-dn.net/?f=f%285%29%20%3D%202%5E%7B5-1%7D%20%3D%202%5E%7B4%7D%20%3D16)
Therefore:
![n_{4} = 8\\ n_{5} = 16](https://tex.z-dn.net/?f=n_%7B4%7D%20%3D%208%5C%5C%20n_%7B5%7D%20%3D%2016)
Answer:
Option B is correct.
Rotation matrix = ![\begin{bmatrix} -3.96 \\ -1.13 \end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D%20-3.96%20%5C%5C%20-1.13%20%5Cend%7Bbmatrix%7D)
Step-by-step explanation:
Given a vector :
, rotation by
radian.
A rotation matrix is a matrix that is used to perform a rotation in Euclidean space.
The standard rotation matrix is given by;
R = ![\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D%5Ccos%20%5Ctheta%20%26%20-%5Csin%20%5Ctheta%20%5C%5C%20%5Csin%20%5Ctheta%20%26%20%5Ccos%20%5Ctheta%20%5Cend%7Bbmatrix%7D)
Then, the matrix of rotation by
radian is:
=
![\begin{bmatrix}x \\ y\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7Dx%20%5C%5C%20y%5Cend%7Bbmatrix%7D)
Then; substitute ![\theta = 120^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%20%3D%20120%5E%7B%5Ccirc%7D)
![\begin{bmatrix}x' \\ y'\end{bmatrix}= \begin{bmatrix}\cos 120^{\circ} & -\sin 120^{\circ} \\ \sin 120^{\circ} & \cos 120^{\circ}\end{bmatrix}\begin{bmatrix}1 \\ 4 \end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7Dx%27%20%5C%5C%20y%27%5Cend%7Bbmatrix%7D%3D%20%5Cbegin%7Bbmatrix%7D%5Ccos%20120%5E%7B%5Ccirc%7D%20%26%20-%5Csin%20120%5E%7B%5Ccirc%7D%20%5C%5C%20%5Csin%20120%5E%7B%5Ccirc%7D%20%26%20%5Ccos%20120%5E%7B%5Ccirc%7D%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D1%20%5C%5C%204%20%5Cend%7Bbmatrix%7D)
or
![\begin{bmatrix}x' \\ y'\end{bmatrix}= \begin{bmatrix} -0.5 & -0.866 \\ 0.866 & -0.5 \end{bmatrix}\begin{bmatrix}1 \\ 4 \end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7Dx%27%20%5C%5C%20y%27%5Cend%7Bbmatrix%7D%3D%20%5Cbegin%7Bbmatrix%7D%20-0.5%20%26%20-0.866%20%5C%5C%200.866%20%26%20-0.5%20%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D1%20%5C%5C%204%20%5Cend%7Bbmatrix%7D)
or
![\begin{bmatrix}x' \\ y'\end{bmatrix}= \begin{bmatrix} -0.5 +4(-0.866) \\ 0.866+4(-0.5)\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7Dx%27%20%5C%5C%20y%27%5Cend%7Bbmatrix%7D%3D%20%5Cbegin%7Bbmatrix%7D%20-0.5%20%2B4%28-0.866%29%20%5C%5C%200.866%2B4%28-0.5%29%5Cend%7Bbmatrix%7D)
Simplify:
![\begin{bmatrix}x' \\ y'\end{bmatrix} = \begin{bmatrix} -3.96 \\ -1.13 \end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7Dx%27%20%5C%5C%20y%27%5Cend%7Bbmatrix%7D%20%3D%20%5Cbegin%7Bbmatrix%7D%20-3.96%20%5C%5C%20-1.13%20%5Cend%7Bbmatrix%7D)
Therefore, the rotation matrix of a given vector is, ![\begin{bmatrix} -3.96 \\ -1.13 \end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D%20-3.96%20%5C%5C%20-1.13%20%5Cend%7Bbmatrix%7D)
Answer:
27x - 8x
Step-by-step explanation:
3x³ - 2x² =
27x - 8x