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On Wikipedia, the terms Exponential Growth and Geometric Growth are listed as synonymous, and defined as when the growth rate of the value of a mathematical function is proportional to the function's current value but I question whether one term is more mathematically correct than the other? For example, there is a Geometric Progression but no Exponential Progression article on Wikipedia, so perhaps the term Geometric is a bit more accurate, mathematically speaking? Why are there two terms for this type of growth? Perhaps exponential growth is more popular in common parlance, and geometric in mathematical circles?

In my 50 or so years of studying mathematics, I've never encountered "geometric growth", but often have met "exponential growth". So that's one small bit of evidence that if you want to sound like most mathematicians, you should use "exponential growth." I wouldn't say either is "more mathematically correct," for the assignment of words as names for concepts is not one that's subject to mathematical evaluation. (Is "continuous" really the right word for the thing we call continuity? Perhaps not...consider a function like
$$
f(x)= begin{cases}
frac{1}{2^k} & text{ if $x = frac{p}{2^k}$ in lowest terms} \
0 & text{otherwise}
end{cases}
$$
which is continuous at every irrational, and at no rational. But if you used any term other than "continuous" for this thing, mathematicians would regard it as somewhere between "peculiar" and "redundant" and "wrong" :) . )

NO.

Geometric and exponential growth are different.

The exponent in geometric sequence formula is always integer.
Hence if you plot the sequence you get step-function kind of discrete plot with sudden jumps.

The exponent of exponential growth is real number. So we have differential (smooth) and continuous plot for the exponential growth.

In compound interest problem, for the finite number of compounding periods, the plot is discrete and it is geometric growth (not continuous)
But if you compound interest continuously (infinite number of compounding periods), you get exponential e in the formula and the growth is exponential.

Look at https://en.wikipedia.org/wiki/File:Compound_Interest_with_Varying_Frequencies.svg

For further info visit: http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

FROM MATHONTHEWEB.COM

12.3 - Exponential Functions

Exponential functions are closely related to geometric sequences. A geometric sequence of numbers is one in which each successive number of the sequence is obtained by multiplying the previous number by a fixed factor $m$. An example is the sequence ${1, 3, 9, 27, 81, dots }$. If we label the numbers in the sequence as ${y_0, y_1, y_2, dots }$ then their values are given by the formula:
$$y_n = y_0 m^n.$$
The geometric sequence is completely described by giving its starting value $y_0 $and the multiplication factor $m$. For the above example $y_0 = 1$ and $m = 3$. Another example is the geometric sequence ${40, 20, 10, 5, 2.5, dots }$ for which $y_0 = 40$ and $m = 0.5$.

The exponential function is simply the generalization of the geometric sequence in which the counting integer n is replaced by the real variable $x$. We define an exponential function to be any function of the form:
$$y = y_0 m^x.$$
It gets its name from the fact that the variable $x$ is in the exponent. The “starting value” $y_0$ may be any real constant but the base $m$ must be a positive real constant to avoid taking roots of negative numbers.

The exponential function $y = y_0 m^x$ has these two properties:
When $x = 0$ then $y = y_0$.

When $x$ is increased by $1$ then $y$ is multiplied by a factor of $m$. This is true for any real value of $x$, not just integer values of $x$. To prove this suppose that $y$ has some value $y_a$ when $x$ has some value $x_a$ . That is:

Now increase $x$ from $x_a$ to $x_{a}+1$. We get:

We see that $y$ is now $m$ times its previous value of $y_a$. If the multiplication factor $m > 1$ then we say that $y$ grows exponentially, and if $m < 1$ then we say that $y$ decays exponentially.