There are taxation situations where the taxable amount includes the tax calculated on the taxable amount (e.g. this is a recursive calculation, as follows)...
Iteration Taxable Amount Tax per iteration
0 $100,000,000.00 $5,000,000.00
1 $105,000,000.00 $250,000.00
2 $105,250,000.00 $12,500.00
3 $105,262,500.00 $625.00
4 $105,263,125.00 $31.25
5 $105,263,156.25 $1.56
6 $105,263,157.81 $0.08
7 $105,263,157.89 $0.00
8 $105,263,157.89 $0.00
9 $105,263,157.89 $0.00
10 $105,263,157.89 $0.00
Tax Rate 5.00%
Effective Tax Rate 5.26%
I would like to determine the Effective Tax Rate without the need to apply the calculation recursively - because no matter what the starting taxable amount the Effective Tax Rate is always the same.
https://www.facebook.com/download/353721708063956/TaxOnTax.xlsx
What you are looking at is a geometric series. Let's say the tax rate is $x$ and the principal $p$. Then what you are doing is computing
$$p+ p x + p x^2 + \cdots = p \sum_{k=0}^{\infty} x^k$$
You may recognize the sum of the geometric series, which may be written simply as
$$p' = \text{net taxable amount} = p \sum_{k=0}^{\infty} x^k = \frac{p}{1-x}$$
The effective tax rate is then
$$\frac{p'-p}{p} = \frac{1}{1-x} - 1 = \frac{x}{1-x} $$
With $x=0.05$, the effective tax rate is about $0.0526$, which agrees with your spreadsheet.