Question

Consider the the formula for finding the total amount of money generated by an account receiving compound interest a = 200 * 1 + 0.08 / 1 when comparing to y = a * 1 + r the general formula for exponential growth which part of the compound interest formula compares best with the variable r

Answer 1

In the context of your question, let's clarify the two formulas you've provided:

1. The compound interest formula you provided seems to be an incomplete expression. The standard formula for compound interest after one compounding period is typically represented as:

\[ a = P \left(1 + \frac{r}{n}\right)^{nt} \]

where:
- \( a \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.

2. The general formula for exponential growth you mentioned is:

\[ y = a \cdot (1 + r) \]

In this context, \( a \) is the initial amount, and \( r \) is the growth rate.

Now, if we focus on the part of the compound interest formula that compares best with the variable \( r \) from the exponential growth formula, we consider the expression inside the parentheses of the compound interest formula.

In the formula:

\[ a = P \left(1 + \frac{r}{n}\right)^{nt} \]

The term \( \frac{r}{n} \) can be related to the growth rate \( r \) in the exponential growth formula, particularly when compounded once per year (\( n = 1 \)). For simplicity, if we assume \( n = 1 \) (annual compounding), then \( r \) from the compound interest formula can be directly compared to \( r \) in the exponential growth formula. In this case:

\[ \frac{r}{1} = r \]

So, when interest is compounded annually, the rate \( r \) in the compound interest formula corresponds directly to the \( r \) in the general exponential growth formula. Therefore, when comparing these, \( r \) in the context of compound interest represents the percentage increase per timeframe.