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

Verified step by step guidance

1. Identify the general term of the sequence given by the formula: \(a_n = \frac{(-1)^{n+1}}{2^n - 1}\).
2. Understand that to find the first four terms, you need to substitute \(n = 1, 2, 3,\) and \(4\) into the formula separately.
3. Calculate each term by plugging in the values of \(n\): for each term, compute the numerator \((-1)^{n+1}\) and the denominator \$2^n - 1$.
4. Write each term as a fraction with the calculated numerator and denominator for \(n=1, 2, 3,\) and \(4\).
5. List the four terms in order: \(a_1, a_2, a_3,\) and \(a_4\) to complete the sequence.

Key Concepts

Sequences and General Terms

A sequence is an ordered list of numbers defined by a general term formula a_n, which gives the nth term. Understanding how to substitute values of n into the formula allows you to find specific terms in the sequence.

Exponents and Powers

Exponents represent repeated multiplication, such as 2^n meaning 2 multiplied by itself n times. Evaluating powers correctly is essential when calculating terms involving expressions like 2^n in the denominator.

Alternating Signs Using (-1)^{n+1}

The factor (-1)^{n+1} causes the terms to alternate in sign because (-1) raised to an even power is positive, and to an odd power is negative. This pattern affects the sign of each term in the sequence.