Answer:
bruh please join back im a new fnaf fandom and i need some help to understand what that is
Explanation:
Answer:
Indirectl Speech:
He said that he is learning French these days•
- Tab ( Complete omitted commands and parameters )
- Space bar ( Display the next screen )
- Up arrow ( Scroll backward through previously entered commands )
- ? ( Provides context-sensitive help )
- Ctrl Shift 6 ( Cancel commands such as trace and ping).
What is CLI tools ?
A command line interface (CLI) is a text-based user interface (UI) used to run programs, manage computer files, and interact with a computer. A command line interface is also known as a command line user interface, console user interface, or character user interface.
Why do we use CLI tools ?
The CLI is a command line utility that accepts text input and executes operating system functions. In the 1960s, this was the only way to interact with a computer, as only computer terminals were in use. In the 1970s and 1980s, command line input was widely used on Unix systems and personal computer systems such as MS-DOS and Apple DOS.
To know more about CLI Tools visit here:
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Answer:
We have proven that the property of the Fibonacci sequence ![[F(n+1)]^2 =[F(n)]^2+F(n-1)F(n+2)](https://tex.z-dn.net/?f=%5BF%28n%2B1%29%5D%5E2%20%3D%5BF%28n%29%5D%5E2%2BF%28n-1%29F%28n%2B2%29)
holds by Mathematical Induction.
Explanation:
For n≥2. We prove that it holds for n=2, Assume that it holds for n=k and prove that it holds for n=k+1
F(1)=1
F(2) = 1
F(3) = F(3 − 1) + F(3 − 2)= F(2) + F(1)=1+1=2
F(4) = F(4 − 1) + F(4 − 2)= F(3) + F(2)=2+1=3
In
We prove it is true for n=2.
When n=2
![L.H.S: [F(n+1)]^2 = [F(2+1)]^2 = [F(3)]^2 = 2^2 =4\\R.H.S: [F(n)]^2+F(n-1)F(n+2) \\=[F(2)]^2+F(2-1)F(2+2)\\= [F(2)]^2+F(1)F(4)\\=1^2+1*3=1+3=4](https://tex.z-dn.net/?f=L.H.S%3A%20%5BF%28n%2B1%29%5D%5E2%20%3D%20%5BF%282%2B1%29%5D%5E2%20%3D%20%5BF%283%29%5D%5E2%20%3D%202%5E2%20%3D4%5C%5CR.H.S%3A%20%5BF%28n%29%5D%5E2%2BF%28n-1%29F%28n%2B2%29%20%20%5C%5C%3D%5BF%282%29%5D%5E2%2BF%282-1%29F%282%2B2%29%5C%5C%3D%20%5BF%282%29%5D%5E2%2BF%281%29F%284%29%5C%5C%3D1%5E2%2B1%2A3%3D1%2B3%3D4)
We assume it is true for n=k and prove that it holds for n=k+1.
When n=k+1 in
Substituting n=k+1 in the LHS: ![[F(n+1)]^2](https://tex.z-dn.net/?f=%5BF%28n%2B1%29%5D%5E2)
and applying: F(k+2)=F(k+1)+F(k)
![LHS: [F(k+2)]^2=[F(k+1)+F(k)]^2\\= [F(k+1)]^2+2F(k+1)F(k)+[F(k)]^2\\=[F(k+1)]^2+F(k)[2F(k+1)+F(k)]\\=[F(k+1)]^2+F(k)[F(k+1)+F(k+1)+F(k)]\\=[F(k+1)]^2+F(k)[F(k+1)+F(k+2)]\\=[F(k+1)]^2+F(k)F(k+3)](https://tex.z-dn.net/?f=LHS%3A%20%5BF%28k%2B2%29%5D%5E2%3D%5BF%28k%2B1%29%2BF%28k%29%5D%5E2%5C%5C%3D%20%5BF%28k%2B1%29%5D%5E2%2B2F%28k%2B1%29F%28k%29%2B%5BF%28k%29%5D%5E2%5C%5C%3D%5BF%28k%2B1%29%5D%5E2%2BF%28k%29%5B2F%28k%2B1%29%2BF%28k%29%5D%5C%5C%3D%5BF%28k%2B1%29%5D%5E2%2BF%28k%29%5BF%28k%2B1%29%2BF%28k%2B1%29%2BF%28k%29%5D%5C%5C%3D%5BF%28k%2B1%29%5D%5E2%2BF%28k%29%5BF%28k%2B1%29%2BF%28k%2B2%29%5D%5C%5C%3D%5BF%28k%2B1%29%5D%5E2%2BF%28k%29F%28k%2B3%29)
Substituting n=k+1 in the RHS :![[F(n)]^2+F(n-1)F(n+2)](https://tex.z-dn.net/?f=%5BF%28n%29%5D%5E2%2BF%28n-1%29F%28n%2B2%29)
RHS=![[F(K+1)]^2+F(K+1-1)F(K+1+2)=[F(k+1)]^2+F(k)F(k+3)](https://tex.z-dn.net/?f=%5BF%28K%2B1%29%5D%5E2%2BF%28K%2B1-1%29F%28K%2B1%2B2%29%3D%5BF%28k%2B1%29%5D%5E2%2BF%28k%29F%28k%2B3%29)
Since the LHS=RHS
Therefore, the property is true.
FOR REFERENCE
When n=k in
Substituting n=k in the LHS: and applying: F(k+1)=F(k)+F(k-1)
![LHS: [F(k+1)]^2=[F(k)+F(k-1)]^2\\= [F(k)]^2+2F(k)F(k-1)+[F(k-1)]^2\\=[F(k)]^2+F(k-1)[2F(k)+F(k-1)]\\=[F(k)]^2+F(k-1)[F(k)+F(k)+F(k-1)]\\=[F(k)]^2+F(k-1)[F(k)+F(k+1)]\\=[F(k)]^2+F(k-1)F(k+2)\\Substituting \: n=k \:in\: th\:e RHS :[F(n)]^2+F(n-1)F(n+2)](https://tex.z-dn.net/?f=LHS%3A%20%5BF%28k%2B1%29%5D%5E2%3D%5BF%28k%29%2BF%28k-1%29%5D%5E2%5C%5C%3D%20%5BF%28k%29%5D%5E2%2B2F%28k%29F%28k-1%29%2B%5BF%28k-1%29%5D%5E2%5C%5C%3D%5BF%28k%29%5D%5E2%2BF%28k-1%29%5B2F%28k%29%2BF%28k-1%29%5D%5C%5C%3D%5BF%28k%29%5D%5E2%2BF%28k-1%29%5BF%28k%29%2BF%28k%29%2BF%28k-1%29%5D%5C%5C%3D%5BF%28k%29%5D%5E2%2BF%28k-1%29%5BF%28k%29%2BF%28k%2B1%29%5D%5C%5C%3D%5BF%28k%29%5D%5E2%2BF%28k-1%29F%28k%2B2%29%5C%5CSubstituting%20%5C%3A%20n%3Dk%20%5C%3Ain%5C%3A%20th%5C%3Ae%20RHS%20%3A%5BF%28n%29%5D%5E2%2BF%28n-1%29F%28n%2B2%29)
![RHS=[F(K)]^2+F(K-1)F(K+2)](https://tex.z-dn.net/?f=RHS%3D%5BF%28K%29%5D%5E2%2BF%28K-1%29F%28K%2B2%29)
LHS=RHS
