Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 1 / n!

Verified step by step guidance

1. Understand the given formula for the nth term of the sequence: \(a_{n} = \frac{1}{n!}\), where \(n!\) (n factorial) is the product of all positive integers from 1 to \(n\).
2. Calculate \(a_1\) by substituting \(n=1\) into the formula: \(a_1 = \frac{1}{1!}\).
3. Calculate \(a_2\) by substituting \(n=2\) into the formula: \(a_2 = \frac{1}{2!}\).
4. Calculate \(a_3\) by substituting \(n=3\) into the formula: \(a_3 = \frac{1}{3!}\).
5. Calculate \(a_4\) by substituting \(n=4\) into the formula: \(a_4 = \frac{1}{4!}\).

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule or formula for its terms. Each term is identified by its position n, and the nth term aₙ gives the value at that position. Understanding how to interpret and use the formula for aₙ is essential to find specific terms.

Factorial Function (n!)

The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, 0! = 1. Factorials grow rapidly and are commonly used in sequences and series.

Evaluating Terms of a Sequence

To find specific terms like a₁, a₂, a₃, and a₄, substitute the term number n into the given formula. For aₙ = 1/n!, calculate the factorial of n and then take its reciprocal. This process allows you to generate the first few terms of the sequence explicitly.