# What is the average rate of growth for the following set of data?

The chart shows the losses and gains in an investment over the course of five years. What was the average growth rate over the course of the five years?

Year$\space\space\space$ Percent Growth

1$\space\space\space\space\space\space\space+10\%$

2$\space\space\space\space\space\space\space-3\%$

3$\space\space\space\space\space\space\space-8\%$

4$\space\space\space\space\space\space\space+4\%$

5$\space\space\space\space\space\space\space+6\%$

I thought that this would mean that if you start with a value of $x$, then the each percentage represents an increase in the previous amount. This would mean that the final amount after 5 years is $x\times1.10\times.97\times.92\times1.04\times1.06$ or $1.0822x$

This would mean that the percent increase is $8.22\%$, but the real answer is $1.59\%$. How would this "average" work and how is my work wrong in the first place?

You are correct that at the end of year five the amount is $1.0822$ times larger than the start value. If you want the average annual growth rate, you take the $1/5$ power of this, as this will be the annual growth that when repeated fve times gives a factor $1.0822$. In fact $1.0822^{1/5} \approx 1.01592$