Why does an account with linear growth and a steady increase of $700 each month have a changing relative difference while an account with exponential growth and a steady 3% increase each month have a steady relative difference of 5% each month?

Not sure just what you mean by "relative difference" but if you mean the growth factor, it is because adding a constant amount reduces the ratio of the ending to the starting balance each month.

6+4 = 10; 10/6 = 66% growth
10+4 = 14; 14/10 = 40%
14+4 = 18; 18/14 = 28%
As the balance grows by the same amount, the proportion shrinks

However, if there is a constant growth factor of 3%/month, then the ratio stays the same: 1.03, or 3% growth

By relative difference I mean the change from one month to another divided by the original value. I'm not talking about the growth factor, I understand that, but we are to calculate the relative difference and talk about why with the linear growth it's different but with exponential growth it stays the same.

huh.

with exponential growth, it has to stay the same. The difference each month depends on the beginning balance.

Think of it. If it grows by 5% each month, then you have
(x*1.05 - x)/x = 1.05 - 1 = 0.05

but with constant growth, say, d, you have
(x+d - x)/x = d/x
which gets smaller as x gets bigger, since d is constant.

To understand why the relative difference changes in the case of linear growth and remains steady in the case of exponential growth, let's break down the two scenarios and analyze them individually.

1. Linear Growth and a Steady Increase of $700 each month:
In this scenario, the account experiences linear growth, meaning it increases by a fixed amount ($700) every month. Let's say the initial value of the account is $x. In the first month, the value would be x + $700, in the second month it would be x + $1,400, and so on.

To calculate the relative difference, we need to determine the percentage change between consecutive months. The formula to calculate relative difference is: (new value - old value) / old value.

Let's consider the difference between the second and first month:
Relative difference = (x + $1,400 - x + $700) / (x + $700).

When we simplify this equation, the x terms cancel out and we are left with:
Relative difference = ($700) / ($700 + x).

As you can see, since the relative difference is a ratio of $700 to the account balance (x + $700), it will change every month as the account balance increases. This means that the relative difference will not remain steady during linear growth.

2. Exponential Growth and a Steady 3% Increase each month:
In this scenario, the account experiences exponential growth, meaning it increases by a certain percentage (3%) of its current value each month. Let's say the initial value of the account is $x. In the first month, the value would be x + (0.03 * x), in the second month it would be (x + (0.03 * x)) + (0.03 * (x + (0.03 * x))), and so on.

To calculate the relative difference, we again need to determine the percentage change between consecutive months. However, since the growth is exponential, the formula for relative difference changes. Instead of using the difference between values, we use the quotient of values in this case.

The formula to calculate the relative difference in case of exponential growth is: (new value - old value) / old value * 100.

Let's consider the difference between the second and first month:
Relative difference = ((x + (0.03 * x)) - x) / x * 100.

When we simplify this equation, we get:
Relative difference = 3%.

As you can see, since the relative difference is a fixed percentage (3%) of the account balance (x), it remains constant during exponential growth. This is because the relative difference is not affected by the account balance. Therefore, the relative difference will remain steady at 5% each month during exponential growth.

In summary, the changing relative difference in the case of linear growth occurs because the increase of a fixed amount ($700) becomes relatively smaller as the account balance grows. On the other hand, the steady relative difference in the case of exponential growth occurs because the percentage increase (3%) remains consistent regardless of the account balance.