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
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?
3 answers
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.
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.
1 answer