{NotebookFileName[],DateString[]}

Collatz conjecture

c1[
n_Integer?OddQ
]:=
3n+1

c1[
n_Integer?EvenQ
]:=
n/2

c1[1]:= 1

c1/@Range[10]

Table[
NestWhileList[c1,n,#≠1&],
{n,1,21,2}
]//Column

Fibonacci/@Range[10]

Sequences

Block[{f=Identity,nmin=1,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(-#&),nmin=1,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->(1/n)),nmin=1,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->-(1/n)),nmin=1,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->Log[(1/(2^n))]),nmin=0,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->Log[(1/n)]),nmin=1,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->Log[(1/(n+1))]-Log[(1/n)]),nmin=1,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->(Sqrt[n])),nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->(Sqrt[n+1])-(Sqrt[n])),nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->((2n+5)/(n+1))),nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

((2n+5)/(n+1))==2+(3/(n+1))//Simplify

2-(3/(n+1))//Together

Block[{f=(n|->((2n-1)/(n+1))),nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->((n-(n^2))/(n+5))),nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

((n-(n^2))/(n+5))+((n(n+5))/(n+5))-n=((n-(n^2)+(n^2)+5n)/(n+5))-n
=((6n)/(n+5))-n
=((6n-6(n+5))/(n+5))+6-n
=((-30)/(n+5))+6-n

((n-(n^2))/(n+5))==((-30)/(n+5))+6-n//Simplify

Assuming[
\[RawEscape]{n>X+6, X>0},
Simplify[-((n-(n^2))/(n+5))>X,n>X+6&&X>0]
]

Simplify[-(((-30)/(n+5))+6-n)>X,n>X+6&&X>0]

Simplify[-(6-n)>X,n>X+6]

Block[{f=Fibonacci,nmin=1,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f=(n|->((Fibonacci[n+1])/(Fibonacci[n]))),nmin=1,nmax=10},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f,b,nmin=0,nmax=10},
b=(x|->(1/2)(x+(2/x)));
f=(n|->Nest[b,1,n]);
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{f,b,nmin=0,nmax=20},
b=(x|->Abs[Log[x]]);
f=(n|->Nest[b,0.5,n]);
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{a=3.1,f,b,nmin=0,nmax=10},
b=(x|->a x(1-x));
f=(n|->Nest[b,0.5,n]);
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{a=2.5,f,b,nmin=0,nmax=10},
b=(x|->a x(1-x));
f=(n|->Nest[b,0.5,n]);
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{a=3.51,f,b,nmin=0,nmax=20},
b=(x|->a x(1-x));
f=(n|->Nest[b,0.5,n]);
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{a=3.6,f,b,nmin=0,nmax=20},
b=(x|->a x(1-x));
f=(n|->Nest[b,0.5,n]);
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{a=3.6,f=Cos,b,nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

Block[{a=3.6,f=(n|->n Cos[n]),b,nmin=0,nmax=20},
ListPlot[f/@Range[nmin,nmax]
,DataRange->{nmin,nmax}
,PlotRange->{{0,nmax+0.2},Full}
,PlotStyle->PointSize[Large]
,Joined->True
,Mesh->All
,MeshStyle->PointSize[Large]
,TicksStyle->Directive[Thick,Bold,FontFamily->"Arial",FontSize->14]
]
]

ln(n+1)-ln(n)

Simplify[(E^x)≥1+x,x∈Reals]

Plot[{
(E^x),x+1
},{x,-2,2}
,PlotRange->{Full,{0,All}}
,PlotLegends->"Expressions"
]