What formula would I use to average (weight) these 10 numbers so that
the 2nd number is 10% greater than the 1st, the 3rd number is 10%
greater than the 2nd, the 4th number is 10% greater than the 3rd, etc
. . . and the total of 10 numbers will equal 200?

Here's one way, starting with a new worksheet.

In A2, put
=A1*1.1

Select A2 and extend down to A10

In A11 put
=SUM(A1:A10)

Here comes the interesting part, where we get a total of 200.

Use
Tools Goal seek
and in the dialog box that this brings up, put
Set cell: A11
To value: 200
By changing: A1
Click "OK" and "OK"

The result is in A1:A10.