How do I calculate equally growing iteration equally between two numbers? I apologize if I do not have the terms correct. I am an artist and I am trying to calculate growth iterations between two numbers to insure correct mathematical growth in the design. 
An Example may be I have an line $9$ inches long. I'd like to draw circles on this line starting with a small circle and ending with a larger circle. How do I calculate this to ensure each circle is larger by multiplying the size for each iteration?
please view PHOTO for reference. NOTE -  The shape can be anything it is the distance on the given line that is important. 
Thanks in advance.
 A: As you say the repeated figure will not always be a circle, things change a little. The key point is that there will be a center point, which need not be visible in the final design. Draw your rightmost figure, then draw the figure immediately to its left, a bit smaller. Now draw in construction lines, a tangent line along the top, I used green, and a tangent line along the bottom, blue. The green, red, and blue lines meet at a point, this is the center of projection. 
Along any one of the lines, the ratio of measured distances from the center gives the overall ratio $r.$ I used millimeters, the ratio here is $r \approx 215 / 265 \approx 0.811.$ Taking any straight line through the center, the distances to matching points in the first two figures should always be $r.$ If not, the figures are not really proportional. 
To get the third figure, multiply everything in the second figure by $r$ again along any straight line through the center. This shows where the third figure goes. Keep drawing new figures until you have approximately your nine inches. 
Do this enough times, either you will know how to do it yourself, or you will be able to ask a question with a more specific answer. 
Mathematically, as Ross indicated, the initial setup is a fixed center point and one figure. Choose a fixed ratio $r$ and draw the second figure. If that is not what you want (I think you want the figures to touch) adjust $r.$ 

A: If the diameter is multiplied by a common factor you have a geometric series.  If the first diameter is $a$, the ratio is $r$, and there are $n$ circles, the total length is $a\frac{r^n-1}{r-1}$  Choose the constants to make this the desired length.
A: Do you want the circles to be proportioned such that the blue line touches each circle at one point, and the green line touches each circle once?

A: Consider the case for circles. It can be shown easily that the radius of consecutive circles form a geometric progression or GP. Let the radius of circles be $r_i$ where $i=1, 2, \cdots, N$. There are four variables:  


*

*Length of line, $L$  (given)

*Number of circles, $N$  

*Radius of first (smallest) circle, $a\; (<\frac L{2N})$ 

*Common ratio of GP, $\rho\; (>1)$


Hence
$$\begin{align}
L&=2(r_1+r_2+r_3+\cdots+r_N)\\
&=2a(1+\rho+\rho^2+\cdots+\rho^{N-1})\\
&=\frac {2a(\rho^N-1)}{\rho-1}\end{align}$$
Using the example provided we have $L=9, N=5$. Assuming we want $a=0.5$ ($<\frac L{2N}=0.9$, so OK), we have to find $\rho$ such that 
$$\frac {\rho^5-1}{\rho-1}=\frac 9{10}$$
Solving numerically gives $\rho=2.24497$. 
So there are $5$ circles with radius starting from $0.1$, and the radius of a given circle is $2.24497$ times the previous smaller one. 
