## Key Ideas

• A savings account which earns compound interest is growing geometrically. At the end of the first year, the initial balance, or principal, is increased by the interest payment. Each successive year, the new balance is the previous balance plus interest, paid as a fixed percentage of that balance. The value of the savings account is determined by the principal (or initial balance), the rate of interest, and the compounding period.
• The interest rate of a savings account for a specified length of time is called the nominal rate. With simple interest, the actual realized percentage is higher and is called the effective rate.
• The nominal rate i for a period during which no compounding is done is given by i= r/m ,where r is the nominal annual rate and m is the number of times interest is compounded per year.
• The compound interest formula for the value of a savings account after n compounding periods is as follows: A = P(1+i)n (i is the interest rate per compounding period).
• The compound interest formula for an annual rate is as follows: A = P(1+r/m)mt (r is the nominal annual rate compounding m times per year for t years).
• If a population is experiencing geometric (exponential) growth, then it is increasing or decreasing by a fixed proportion of its current value with each measurement. The proportion is called the growth rate of the population.
• Accounts earning compound interest will grow more rapidly than accounts earning simple interest. In general, geometric growth (such as compound interest) is much more dramatic than arithmetic growth (such as simple interest).
• For a nominal interest rate r compounded m times per year, the effective annual rate of interest, or APY, is (1+r/m)m -1.
• Given the formula A = P(1+i)n we may wish to find i for some applications. To do this we calculate i = (A/P)1/n - 1.