In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a40 when a1 = 1000, r = - 1/2

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of sequence can be expressed in the form a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

General Term Formula

The general term formula for a geometric sequence allows us to calculate any term in the sequence based on its position. It is given by a_n = a_1 * r^(n-1). This formula is essential for finding specific terms, such as the 40th term in the sequence, by substituting the values of a_1, r, and n.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is denoted by r and can be positive or negative. In the given problem, the common ratio is -1/2, indicating that each term will alternate in sign and decrease in magnitude by half, which is crucial for accurately calculating the specified term.