Algebra and Trigonometry, 5th Edition

11.1 Sequences and Series

• Find terms of sequences given the nth term.

• Look for a pattern in a sequence and try to determine a general term.

• Convert between sigma notation and other notation for a series.

• Construct the terms of a recursively defined sequence.

In this section, we discuss sets or lists of numbers, considered in order, and their sums.

Sequences

Suppose that $1000 is invested at 4%, compounded annually. The amounts to which the account will grow after 1 year, 2 years, 3 years, 4 years, and so on, form the following sequence of numbers:

$$\begin{array}{cccc}\mathbf{(}\mathbf{1}\mathbf{)}& \mathbf{(}\mathbf{2}\mathbf{)}& \mathbf{(}\mathbf{3}\mathbf{)}& \mathbf{(}\mathbf{4}\mathbf{)}\\ \downarrow & \downarrow & \downarrow & \downarrow \\ \mathrm{\$1040.00}& \mathrm{\$1081.00}& \mathrm{\$1124.86}& \mathrm{\$1169.86.}\end{array}$$

We can think of this as a function that pairs 1 with $1040.00, 2 with $1081.60, 3 with $1124.86, and so on. A sequence is thus a function, where ...