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

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Identify the general term of the sequence given by the formula: \(a_n = \frac{(-1)^{n+1}}{2^n - 1}\).

Understand that to find the first four terms, you need to substitute \(n = 1, 2, 3,\) and \(4\) into the formula separately.

Calculate each term by plugging in the values of \(n\): for each term, compute the numerator \((-1)^{n+1}\) and the denominator \(2^n - 1\).

Write each term as a fraction with the calculated numerator and denominator for \(n=1, 2, 3,\) and \(4\).

List the four terms in order: \(a_1, a_2, a_3,\) and \(a_4\) to complete the sequence.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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