In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)^n+1/(2^n−1)

Verified step by step guidance

1. Identify the general term of the sequence: $a_n = \frac{(-1)^{n+1}}{2^n - 1}$.
2. To find the first term $a_1$, substitute $n = 1$ into the general term: $a_1 = \frac{(-1)^{1+1}}{2^1 - 1}$.
3. To find the second term $a_2$, substitute $n = 2$ into the general term: $a_2 = \frac{(-1)^{2+1}}{2^2 - 1}$.
4. To find the third term $a_3$, substitute $n = 3$ into the general term: $a_3 = \frac{(-1)^{3+1}}{2^3 - 1}$.
5. To find the fourth term $a_4$, substitute $n = 4$ into the general term: $a_4 = \frac{(-1)^{4+1}}{2^4 - 1}$.

Key Concepts

Sequences

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of each term is typically denoted by an index, often starting from 1. Understanding how to identify and generate terms from a given formula is essential for working with sequences.

General Term of a Sequence

The general term of a sequence, often denoted as an, is a formula that defines the nth term of the sequence in terms of n. In this case, the general term is given by an = (−1)^(n+1)/(2^n−1). This formula allows us to calculate any term in the sequence by substituting different values of n, which is crucial for finding specific terms.

Evaluating Expressions

Evaluating expressions involves substituting values into a mathematical formula to compute a result. For sequences, this means plugging in the index values (n = 1, 2, 3, 4, etc.) into the general term formula to find the corresponding terms. Mastery of this skill is necessary to accurately derive the first four terms of the sequence.