If a person is saving for the future and invests $1000 at 5%, compounded annually, the value of the investment 40 years later would be about $7040. Even better, if the interest is compounded daily, which is a common method today, it would be worth about $7390 in 40 years. A person saving for retirement would do much better by putting aside $1000 each year and letting the interest accumulate. If $1000 is invested each year at 5%, compounded annually, the total investment would be worth about $126,840 after 40 years. If the interest is compounded daily, it would be worth about $131,000 in 40 years. Obviously, if more is invested, or the interest rate is higher, the value of these investments would be higher.

Each of these values can be found quickly since the values of the annual investments form what is called a geometric sequence, and formulas can be formed for such sums. Such formulas involving compound interest are widely used in calculating values such as monthly car payments, home mortgages, and annuities.

A sequence is a set of numbers arranged in some specific way and usually follows a pattern. Sequences have been of interest to people for centuries. There are records that date back to at least 1700 b.c.e. showing calculations involving sequences. Euclid, in his Elements, dealt with sequences in about 300 b.c.e. More advanced forms of sequences were used extensively in the study of advanced mathematics in the 1700s and 1800s. These advances in mathematics have been very important in many areas of science and technology.

Of the many types of sequences, we study certain basic ones in this chapter. Included are those used in the expansion of a binomial to a power. We show applications in areas such as physics and chemistry in studying radioactivity, biology in studying population growth, and, of course, in business when calculating interest.