Remove["Global`*"];

pRain[i_]:=Module[{prop},
prop={};
If[i==0,prop=p,AppendTo[prop,α pRain[i-1]+(1-β) (1-pRain[i-1])]];
prop
]

process[n_,pp_,αα_,ββ_]:=Module[{(*r,expr1,expr2,c*)},
r=Table[{i,pRain[i],0},{i,0,n}];
r[[All,3]]=N[r[[All,2]]/.{α->αα,β->ββ,p->pp},16];
(*expr1=Text["Probability it will rain today (day 0) is p="<>ToString[pp],{2,10}];*)
expr2=Text[TableForm[Simplify[r],
TableHeadings->{None,{"N","Symbolic probability of rain at day 'N'","numerical value"}}]];
(*Graphics[{{expr1},{expr2}},ImageSize->{600,140}]*)
Graphics[expr2,ImageSize->{600,140}]
]

s=Manipulate[process[n,p,α,β],{{n,5,"number of days"},0,6,1,Appearance->"Labeled"},{{p,.025,"p=probability rain today"},.0125,1,.05,Appearance->"Labeled"},{{α,.4,"α = P(R1 | R0)"},.1,1,.1,Appearance->"Labeled"},{{β,.3,"β = P((Subscript[(R^c), 1])| (Subscript[(R^c), 0]))"},.1,1,.1,Appearance->"Labeled"},AutorunSequencing->{{2,60}},FrameLabel->"The rain problem, #54 chapter 1, Math 502 probability and Statistics, CSUF. by Nasser Abbasi"]

Export["s2.swf",s]

k[n_]:=Mod[n,2]

Mod[11,2]

Remove["Global`*"]
prop[α_,β_,n_,p_]:=Module[{},
Mod[n,2]+(-1)^(n+1)Floor[n,2]α+Ceiling[n,2]β+(-1)^(n+1)Sum[i,{1,n}]α β+p((-1)^n+(-1)^n n α+(-1)^n n β+(-1)^n ((n^2)-n))]
prop[.1,.4,100,.1]

?Round

prop[α_,β_,p_,n_]:=Module[{},
Mod[n,2]+(-1)^(n)Floor[n/2]α+(-1)^(n) Ceiling[n/2]β+(-1)^(n+1)Sum[i,{i,1,n-1}]α β+p((-1)^n+(-1)^(n+1) n α+((-1)^(n+1)) n β+((-1)^n )((n^2)-n)α β)]

r=Table[{n,prop[.1,.3,.1,n]},{n,100,1000,100}]

prop[.1,.3,.4,30]

prop[α_,β_,n_,p_]:=Module[{},
p((-1)^n+(-1)^(n+1) n α+(-1)^(n+1) n β+(-1)^n (n^2-n))]
prop[.1,.4,7,.1]

Sum[i,{i,4}]

?Sum

i

Table[((n^2)-n)/.n->i,{i,2,6}]

(3 a+3 b-15 a b+p(1-6 a-6 b+30 a b))/.{a->.1,b->.3,p->.5}

prop[α_,β_,n_,p_]:=Module[{},
Floor[n/2]α+ Ceiling[n/2]β-Sum[i,{i,1,n-1}]α β+p(1- n α- n β+((n^2)-n)α β)]
prop[.1,.3,30,.4]

Sum[i,{i,1,6-1}]

?Floor

prop[n_]:=Module[{},
Mod[n,2]+(-1)^(n)Floor[n/2]α+(-1)^(n) Ceiling[n/2]β+(-1)^(n+1)Sum[i,{i,1,n-1}]α β+p((-1)^n+(-1)^(n+1) n α+(-1)^(n+1) n β+(-1)^n ((n^2)-n)α β)]

prop[60]/.{α->.1,β->.3,p->.4}