use exp(...) instead of e^(...)
use parentheses (...) instead of square backets [...] because square brackets are used for ?lists

The most efficient way is to create a sequence S by a for loop

r := 0.8: x := 0.9:  # parameters
S := 1: # initialization of the sequence
for i to 99 do 
  x := r*x*(1-x):# calculate x[i]
  S := S,x       # append to the sequence
end do:
X := [S];  # make a list

You will find it easy to make a procedure of this algorithm.

Firstly, I recommend to use Maple notation for input display to get full control over your input:
Choose: Tools → Options → Display → Input Display: Maple Notation

Now you can consistently use a[1], a[n], etc for indexed variables. These will (nearly!) always be displayed as a₁ , a_n, etc.
But you must realize that by using the names a[1] etc. automatically a ?table named a is created. This is nearly never harmful, except if you want to use the name a (without subscript):

a[1],a[2] := 3,5;
                               

      a₁, a₂:= 3, 5

a := 4: a[1];

      4₁

Another way to use "litaral subscripted" names, is mathML code. Perhaps you will better use a macro as an abbreviation:

macro(a1 = `#msub(mi("a"),mi("1"))`):

Whenever you type a1, the output will be a₁

Many times you use a multinlication sign in an intended function application. For example:

cos*alpha

Mh*(V, alpha)

where

cos(alpha)

Mh(V, alpha)

is meant.

See the ?VectorCalculus package (it uses e_x, e_y, e_z for i, j, k).

If you calculate

S,U,V := SmithForm(A, output=['S','U','V']);

then V is the matriof the column operations (and U the matrix of row operations)

B := <A[..,2..-1]|A[..,1]>;

The problem is that maple tries to evaluate the piecewise function before the variables have a value.
You can avoid this by

f := (x,y,z) -> piecewise( (x>=0 and y>=0 and z>= 0), 2*x+3*y+z, undefined ):
implicitplot3d( f(x,y,z) = 6, x = -3 .. 3, y = -3 .. 3, z = -1 .. 7, color = red, 
   axes = normal, scaling = constrained, style = surface);

(f@op)~(S);

By the way, do you realize the difference between a set and a list? See ?set.
Try for example

f := (x,y) -> x^2+y^2:
S := {[3,4],[4,3],[0,5],[5,0]};  # set

or

S := [[3,4],[4,3],[0,5],[5,0]];   # list

Perhaps more general:

eqs := {x-lambda*p=0, y-lambda=0}:
solve( eqs, {x,y} ): eq := x/y = subs(%, x/y);

Because this isn't true if x<0

a := (x*ln(x)-ln(1/(1+x))-x*ln(x/(1+x)))/((1+x)*ln(1+x)):
simplify(a) assuming x>0;

(If x < 0 the expression is complex)

I am afraid that you have to plot the right angle symbol explicitly by connecting three (calculated) points.

with(plots):
p1 := plot( x, x=-1..3, color=blue ):
a,b := [1,2], [2,1]:  # two points
p2 := plot ( [a,b], thickness= 3 ):
c := 3/2:  # [c,c] is the intersection point
h := 0.1:  # size of the right angle symbol
ra := plot( [[c-h,c+h], [c,c+2*h], [c+h,c+h]], color=black ): 
display( {p1,p2, ra} );

A := Matrix( 12, (i,j) -> (i-1)^(j-1) );

Why not Export the worksheet as HTML or as PDF?

See the "note" in ?sum

 s := x -> piecewise(x = 1, 1, 1 < x, add(s(i), i = 1 .. x-1));

will do what you want.