Question

Determine the \(n^{\text {th }}\) term of the given sequence. $$1,1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots$$

Short answer

The \(n^{\text{th}}\) term is \(a_n = \frac{1}{(n-1)!}\) for \(n \geq 3\), with \(a_1 = 1\) and \(a_2 = 1\).

Step-by-step solution

Observe the Sequence

Look at the given sequence: \(1, 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots\). Notice that after the first two terms, each term is a fraction.

Understand the Pattern

Notice that the denominators in the sequence represent factorials: \(1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120\). This suggests that the denominator of the \(n^{\text{th}}\) term is \((n-1)!\).

Write a General Expression

For the \(n^{\text{th}}\) term \(a_n\), observe that for \(n \geq 3\), \(a_n = \frac{1}{(n-1)!}\).

Validate the Expression

Verify the general expression by checking the first few terms:\- For \(n = 1\), \(a_1 = 1\). It matches our sequence.\- For \(n = 2\), \(a_2 = 1\). It matches our sequence.\- For \(n = 3\), \(a_3 = \frac{1}{2!} = \frac{1}{2}\). It matches our sequence.\- For \(n = 4\), \(a_4 = \frac{1}{3!} = \frac{1}{6}\). It matches our sequence. This pattern holds for subsequent terms as well.

Key concepts

factorials

Factorials are a key concept in mathematics that involve multiplying a series of descending natural numbers. For instance, the factorial of a number \( n \) is denoted by \( n! \) and is calculated as \( n \times (n-1) \times (n-2) \times \ldots \times 1 \). The value of \( 0! \) is defined as 1. This concept is essential in permutations, combinations, and series calculations.

In the context of this sequence, you may notice that the denominators are factorials of the numbers:

• 1! = 1
• 2! = 2
• 3! = 6
• 4! = 24
• 5! = 120

Recognizing factorials helps in understanding many mathematical sequences and enables us to predict or describe the behavior of complex patterns.

general expression

Creating a general expression for a sequence allows us to describe any term without listing all previous terms. This is useful for efficiently finding the \(n^{\text{th}}\) term directly. In the provided sequence, once we notice that the denominators are factorials, we can derive a general expression.

Here, for \( n \geq 3 \), each term \( a_n \) in the sequence is given by the formula \( a_n = \frac{1}{(n-1)!} \). This expression compactly represents the series, saving time and effort especially when dealing with larger sequences. It encapsulates the essence of the sequence’s behavior.

nth term

The concept of the \(n^{\text{th}}\) term is central in sequences and series analysis, as it allows us to identify any specific term in the sequence without sequential computation. For the given sequence, the \(n^{\text{th}}\) term is expressed differently based on its position:

• If \( n = 1 \), \( a_1 = 1 \)
• If \( n = 2 \), \( a_2 = 1 \)
• If \( n \geq 3 \), \( a_n = \frac{1}{(n-1)!} \)

This differentiated approach helps in accurately identifying the term in relation to its position, especially where patterns emerge clearly after initial anomalies, as seen here with the first two terms.

sequence analysis

Sequence analysis involves examining a series of numbers or terms to uncover patterns or rules governing their generation. It is like being a detective, hunting for regularities that simplify predictions or calculations.

In the given sequence, through analysis, we found that the pattern emerges after the initial terms, with factorials playing a key role in the denominators of each term beyond the second. This understanding allows for the formulation of a reliable expression for any subsequent terms.

Typically, analyzing sequences helps in:

• Identifying recurring elements
• Predicting future terms
• Simplifying complex calculations
• Deriving new mathematical properties

This makes sequence analysis a valuable tool in both theoretical and applied mathematics.