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

Verified step by step guidance

1. Identify the general term of the sequence given by the formula \(a_n = \frac{(-1)^n}{2^n}\), where \(n\) is the term number starting from 1.
2. Calculate the first term \(a_1\) by substituting \(n=1\) into the formula: \(a_1 = \frac{(-1)^1}{2^1}\).
3. Calculate the second term \(a_2\) by substituting \(n=2\) into the formula: \(a_2 = \frac{(-1)^2}{2^2}\).
4. Calculate the third term \(a_3\) by substituting \(n=3\) into the formula: \(a_3 = \frac{(-1)^3}{2^3}\).
5. Calculate the fourth term \(a_4\) by substituting \(n=4\) into the formula: \(a_4 = \frac{(-1)^4}{2^4}\).

Key Concepts

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 from the explicit formula is essential for writing out the sequence.

Explicit Formula for Sequences

An explicit formula directly defines the nth term of a sequence as a function of n, allowing calculation of any term without knowing previous terms. For example, aₙ = (-1)ⁿ / 2ⁿ gives a direct way to compute each term by substituting n.

Exponentiation and Alternating Signs

Exponentiation involves raising a base to a power, such as 2ⁿ. The term (-1)ⁿ introduces alternating signs 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.