13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)

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Identify the given explicit formula for the sequence: \(a_n = \frac{2^{n+1}}{2^n + 1}\).

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

Calculate the first term by substituting \(n=1\): \(a_1 = \frac{2^{1+1}}{2^1 + 1} = \frac{2^2}{2 + 1}\).

Calculate the second term by substituting \(n=2\): \(a_2 = \frac{2^{2+1}}{2^2 + 1} = \frac{2^3}{4 + 1}\).

Similarly, calculate the third and fourth terms by substituting \(n=3\) and \(n=4\) into the formula, respectively.

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주요 개념

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

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule or formula. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how to find terms using the explicit formula is essential for generating the sequence.

Explicit Formula for Sequences

An explicit formula expresses the nth term of a sequence directly in terms of n, allowing calculation of any term without knowing previous terms. For example, aₙ = (2ⁿ⁺¹) / (2ⁿ + 1) gives a direct way to find the nth term by substituting n.

Evaluating Exponential Expressions

Evaluating terms in the sequence requires calculating powers of 2, such as 2ⁿ and 2ⁿ⁺¹. Understanding how to compute and simplify exponential expressions is crucial to accurately determine each term's value.

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)