The steps in the patterns A and B form a sequence
The attached diagram represents dots in patterns A and B
<h3>How to draw the diagrams</h3>
For pattern A, we have:
Step 2 = 10 dots
Increment of 3 dots in each step
This means that:
Step 0 = 4 dots
Step 1 = 7 dots
Step 2 = 10 dots
Step 3 = 13 dots
For pattern A, we have the following rule
![T_n =2n^2 + 1](https://tex.z-dn.net/?f=T_n%20%3D2n%5E2%20%2B%201)
When n = 0 to 3, we have:
![T_0 =2(0)^2 + 1 =1](https://tex.z-dn.net/?f=T_0%20%3D2%280%29%5E2%20%2B%201%20%3D1)
Step 0 = 1 dot
![T_1 =2(1)^2 + 1 =3](https://tex.z-dn.net/?f=T_1%20%3D2%281%29%5E2%20%2B%201%20%3D3)
Step 1 = 3 dots
![T_2 =2(2)^2 + 1 =9](https://tex.z-dn.net/?f=T_2%20%3D2%282%29%5E2%20%2B%201%20%3D9)
Step 2 = 9 dots
![T_3 =2(3)^2 + 1 =19](https://tex.z-dn.net/?f=T_3%20%3D2%283%29%5E2%20%2B%201%20%3D19)
Step 3 = 19 dots
Next, we draw the diagrams of both patterns (see attachment)
Read more about patterns at:
brainly.com/question/15590116
Answer:
![x = 24](https://tex.z-dn.net/?f=x%20%3D%2024)
Step-by-step explanation:
![- x = - 24](https://tex.z-dn.net/?f=%20-%20x%20%3D%20%20-%2024)
If you didn't know, this equation is actually equal to
![- 1 \times x = - 1 \times 24](https://tex.z-dn.net/?f=%20-%201%20%5Ctimes%20x%20%3D%20%20-%201%20%5Ctimes%2024)
Divide -1 on both sides:
![x = 24](https://tex.z-dn.net/?f=x%20%3D%2024)
Suppose
is another solution. Then
![\begin{cases}y_2=vx^3\\{y_2}'=v'x^3+3vx^2//{y_2}''=v''x^3+6v'x^2+6vx\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dy_2%3Dvx%5E3%5C%5C%7By_2%7D%27%3Dv%27x%5E3%2B3vx%5E2%2F%2F%7By_2%7D%27%27%3Dv%27%27x%5E3%2B6v%27x%5E2%2B6vx%5Cend%7Bcases%7D)
Substituting these derivatives into the ODE gives
![x^2(v''x^3+6v'x^2+6vx)-x(v'x^3+3vx^2)-3vx^3=0](https://tex.z-dn.net/?f=x%5E2%28v%27%27x%5E3%2B6v%27x%5E2%2B6vx%29-x%28v%27x%5E3%2B3vx%5E2%29-3vx%5E3%3D0)
![x^5v''+5x^4v'=0](https://tex.z-dn.net/?f=x%5E5v%27%27%2B5x%5E4v%27%3D0)
Let
, so that
![\begin{cases}u=v'\\u'=v''\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Du%3Dv%27%5C%5Cu%27%3Dv%27%27%5Cend%7Bcases%7D)
Then the ODE becomes
![x^5u'+5x^4u=0](https://tex.z-dn.net/?f=x%5E5u%27%2B5x%5E4u%3D0)
and we can condense the left hand side as a derivative of a product,
![\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E5u%5D%3D0)
Integrate both sides with respect to
:
![\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E5u%5D%5C%2C%5Cmathrm%20dx%3DC)
![x^5u=C\implies u=Cx^{-5}](https://tex.z-dn.net/?f=x%5E5u%3DC%5Cimplies%20u%3DCx%5E%7B-5%7D)
Solve for
:
![v'=Cx^{-5}\implies v=-\dfrac{C_1}4x^{-4}+C_2](https://tex.z-dn.net/?f=v%27%3DCx%5E%7B-5%7D%5Cimplies%20v%3D-%5Cdfrac%7BC_1%7D4x%5E%7B-4%7D%2BC_2)
Solve for
:
![\dfrac{y_2}{x^3}=-\dfrac{C_1}4x^{-4}+C_2\implies y_2=C_2x^3-\dfrac{C_1}{4x}](https://tex.z-dn.net/?f=%5Cdfrac%7By_2%7D%7Bx%5E3%7D%3D-%5Cdfrac%7BC_1%7D4x%5E%7B-4%7D%2BC_2%5Cimplies%20y_2%3DC_2x%5E3-%5Cdfrac%7BC_1%7D%7B4x%7D)
So another linearly independent solution is
.
Answer:
4
Step-by-step explanation:
25+25=50
50*2=100
100/25=4