I am trying to get the average rate of growth across a series of numbers that will lead to the same sum as the discrete rates of growth.
For example, if I have 1 unit at $t=1$ that grows at 2%, then 3%, then 2% again the resulting sum of units would be $$1+1\cdot(1.02)+1\cdot(1.02)\cdot(1.03)+1\cdot(1.02)\cdot(1.03)\cdot(1.02)=4.142212$$
Unless I'm making a calculative error, I've found that the arithmetic, geometric, and harmonic averages all lead to an incorrect sum when scaled linearly over the same 4 periods. Is there a method to find the correct average growth rate that will give the same sum?
First time question asker, please forgive the potentially convoluted question. Welcome any tips on better question formatting.
Let's introduce some notation to describe your question. Suppose we perform $N$ measurements of a time-dependent quantity $x(t)$. Denote the observed value of $x$ at time $t_i$ as $x_i=x(t_i)$ for $i \in \{1,2,...,N\}$. Then define $r_{i+1}=\frac{x_{i+1}}{x_i}$ as the fractional increase of $x$ between times $t_i$ and $t_{i+1}$ for $i \in \{1,2,...,N-1\}$. Moreover, it is convenient to assign $r_1=1$. It follows by iterative application of this definition that $x_i=x_1 \prod_{j=1}^{i} r_j$, for all $i \in \{1,2,...,N\}$. The (unweighted) average of $x$ across the $N$ measurements is $$\bar{x}=\frac{1}{N} \sum_{i=1}^{N} x_i=\frac{x_1}{N} \sum_{i=1}^{N} \prod_{j=1}^{i} r_j\tag{1}.$$ The metric you are considering for the average fractional change is then $$\bar{r}=\frac{\bar{x}}{x_1}=\frac{1}{N} \sum_{i=1}^{N} \prod_{j=1}^{i} r_j \tag{2}.$$ For computational purposes, it is useful to note that $\sum_{i=1}^{N} \prod_{j=1}^{i} r_j =r_1\left(1+r_2\left(1+...r_{N-1}\left(1+r_N\right)\right)\right)$. In terms of your example, we would have that $N=4$, $x_1 = 1$, and $\vec{r}=(1,1.02,1.03,1.02)$. Then we would calculate $\bar{r}=1+r_2(1+r_3(1+r_4))=1.035553$ and $\bar{x}=\bar{r}x_1=1.035553$. It should hopefully be clear from $(2)$ why $\bar{r}$ is not simply the arithmetic, geometric, or harmonic mean of the components of $\vec{r}$.
In fact, I would argue that $\bar{r}$ as defined in $(2)$ should not be understood as an average of the components of $\vec{r}$. The prototypical property of an averaging function $\bar{s}:S^N \to S$ for $N$ elements of a set $S$ is that for all $s \in S$, we have that $\bar{s}(s,s,...,s)=s$. In the case that $S=(0,\infty)$ and $r_i=r \in S$ for all $i \in \{1,2,...N\}$, $$\bar{r}=\frac{1}{N} \sum_{i=1}^{N} \prod_{j=1}^{i} r = \frac{1}{N} \sum_{i=1}^{N} r^i = \frac{r}{N} \lim\limits_{r' \to r} \frac{1-r'^N}{1-r'},$$ where we have used the standard formula for a finite geometric series (the limit being needed for when $r=1$ or $N=1$). For all $r \neq 1$ and $N\geq 2$, it is then apparent that $\bar{r} = \frac{r}{N}\frac{1-r^N}{1-r} \neq r$, and even more alarming, $\lim\limits_{r \to \infty} \frac{\bar{r}}{r} = \frac{1}{N} \lim\limits_{r \to \infty} \frac{1-r^N}{1-r}=\infty$.
I would instead think of $\bar{r}$ as the fractional change between the first measurement of $x$ and the sample mean $\bar{x}$ as it is defined in $(2)$. I am afraid we do not have reason to extend its meaning beyond this.