Do we know all the different Ns, and are they different? If so, you’re just solving for x in the equation

i1*i2*i3...iN = x^N

For the sake of clarity of the post, I’m defining i as, say, 1.12, not 12%. I know you know about dividing by 100 and adding 1 and doing the reverse with the answer, so I’m not going to clutter up the equation with it.

Okay, let’s fill in numbers. Say the inflation rates for 2000-2004 are 10.5, 10.7, 5.3, 6.4, and 8.7. What is the annualized rate of inflation for the entire period? Is it just the mean?

Yeah, pretty much.
Let’s define some variables. S is the starting amount. E is the ending amount. The various changes are W, X, Y, and Z, in the decimal form, not percentage. That is, 1.105 not 10.5%.

In reality, the number changes like so:

Swxyz = E

That is, the starting amount gets multiplied by each change. as you go through each iteration. But you want to know what the average is. You want to know A in the following:

SAAAA = E

So by substitution, we get:

SA^4 = SWXY*Z

Cancelling an S and taking the 4th root of both sides yields:

(WXY*Z)^(1/4) = A

If you have a number of variables other than 4, you can generalize it as:

(i1i2i3…iN) ^ (1/N) = A

You messed up when you tried to arithmetically average the percentages instead of geometrically. In other words, you declared that:

AAA*A = W+X+Y+Z
4A = W+X+Y+Z
A = (W+X+Y+Z) / 4

The error is twofold. One: AAA*A is A^4, not 4A, and Two: W+X+Y+Z never shows up in the equation and is thus a meaningless quantity. They’re originally multiplied, not added.