Problem 15 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\\{n\\}\)

Step-by-Step Solution

Answer

1, 2, 3, 4, 5.

Step 1: Understand the Sequence

The given sequence is \(\right\{ s_{n} \right\}=\right\{n\right\}\right\). This notation means each term in the sequence is equal to its position number (n).

Step 2: Identify the First Term

For the first term, set \(n = 1\). Hence, \(s_1 = 1\).

Step 3: Identify the Second Term

For the second term, set \(n = 2\). Hence, \(s_2 = 2\).

Step 4: Identify the Third Term

For the third term, set \(n = 3\). Hence, \(s_3 = 3\).

Step 5: Identify the Fourth Term

For the fourth term, set \(n = 4\). Hence, \(s_4 = 4\).

Step 6: Identify the Fifth Term

For the fifth term, set \(n = 5\). Hence, \(s_5 = 5\).

Step 7: Final Answer

The first five terms of the sequence \(\right\{ s_{n} \right\}=\right\{n\right\}\right\) are 1, 2, 3, 4, and 5.

Key Concepts

Sequence in Mathematics

A sequence in mathematics is a list of numbers arranged in a specific order. Sequences are essential concepts that appear in various mathematical topics, ranging from basic arithmetic to complex calculus.
An important property of sequences is that they are ordered sets, meaning position and arrangement matter.
Here, we focus on identifying and listing sequence terms.
For example, in the sequence \(\right\{ s_n \right\} \), each term follows a specific rule. Understanding this rule is crucial to determining any term's value.

Terms in a Sequence

Terms are the individual elements or numbers in a sequence. Each term has a specific position or order within the sequence.

• For instance, the first term is often represented as \(s_1\), the second term as \(s_2\), and so on.
• In the sequence \(\right\{ s_n \right\}=\right\{ n \right\}\), each term is equal to its position number.

This implies that:
\(s_1 = 1\)
\(s_2 = 2\)
\(s_3 = 3\)
\(s_4 = 4\)
and \(s_5 = 5\). Listing these terms correctly is key to mastering sequences.

Position Number in a Sequence

The position number in a sequence indicates the location of a term. It is represented by \(n\) in mathematics.

• For the sequence given, \(\right\{ s_n \right\} \), the position number is essential in determining the value of each term.
• When \(n = 1\), the first term is \(s_1 = 1\).
• When \(n = 2\), the second term is \(s_2 = 2\).

Hence, by knowing the position number \(n\), you can identify any term within the sequence, making it an integral part of understanding and solving sequence problems.