If it helps... for any process that moves exponentially (or geometrically), you can remap the values onto a logarithmic scale. This linearizes the process and allows you to think about it more naturally.

To transform the series, simply convert all values to the log of that value. The base of the log doesn't matter, so you can use the natural log (ln) or a base ten log (sometimes log, sometimes log10).

If the value of your stock starts at 100, then increases to 200, then drops back down to 100, that's a percentage change of +100% and -50% for an 'average gain' of 25% and a net result of zero. Very confusing.

But, let's take the natural log of those values. The stock starts at 4.6, it increases to 5.3, and then decreases back to 4.6. In log-space, you don't multiply, you add. So the stock went up 0.7 'nats' and then back down 0.7 'nats' for an average change of 0.

Now we're in units of 'nats'. How the hell is this any more helpful? Well, it lets us calculate back to our normal units and we can get the true average return very easily.

Our stock started out at $100. Ten years later it's worth $750. What's the average return?

1) Convert to nats: Stock went from 4.6 to 6.6 in ten years. That's an average change of 0.2 nats per year.

2) Convert back: the exp() function is the inverse of the natural log. e^0.2 = about 1.221. Your stock averaged a 22.1% gain each year.

This answer will hold no matter what the intermediate values are between where you started and stopped, so you can skip the middle, or you can average the logarithmic values as you are trying to do with the percentage changes.

$100 * 1.221^10 is about $750.