Problem 10 - Algebra and Trigonometry

Question

Find the first four terms and the 100th term of the sequence. \(a_{n}=3\)

Step-by-Step Solution

Answer

All terms including the 100th term are 3.

Step 1: Understanding the Sequence

We have the sequence defined by the formula \( a_n = 3 \). This indicates that each term in the sequence is 3, regardless of the position \( n \). This type of sequence is called a constant sequence.

Step 2: Finding the First Term

To find the first term of the sequence, we substitute \( n = 1 \) into the formula. So, \( a_1 = 3 \).

Step 3: Calculating the Second Term

To find the second term, substitute \( n = 2 \) into the formula. Thus, \( a_2 = 3 \).

Step 4: Determining the Third Term

Similarly, for the third term, substitute \( n = 3 \) into the formula. Therefore, \( a_3 = 3 \).

Step 5: Evaluating the Fourth Term

For the fourth term, let \( n = 4 \) in the formula. Thus, \( a_4 = 3 \).

Step 6: Calculating the 100th Term

To find the 100th term, substitute \( n = 100 \) into the formula. This gives \( a_{100} = 3 \). Since it's a constant sequence, the 100th term is also 3.

Key Concepts

Sequence Notation

In mathematics, sequences are sets of numbers arranged in a specific order. They are denoted using special notation to define their terms. For example, the term "sequence notation" typically involves subscripting to indicate which term in the sequence you are referring to.
For the sequence we have, the notation is given by \( a_n \), where \( n \) represents the position of a term within the sequence. Here, \( a_n = 3 \) implies that each term is 3.

• "\( a \)" is the sequence term.
• "\( n \)" indicates the term's position.

Such notation offers clarity and precision, making it easy to calculate and identify any term in the sequence.

Term Calculation

Calculating terms in a sequence is a straightforward process when a clear formula is provided. To find specific terms, substitute the position number \( n \) into the formula.
Given the sequence formula \( a_n = 3 \), each term calculation will yield the same result because the sequence is constant:

• For the first term, substitute \( n = 1 \) to get \( a_1 = 3 \).
• For the second term, substitute \( n = 2 \) and find \( a_2 = 3 \).
• This process repeats identically for any position \( n \).
• For the 100th term, even at \( n = 100 \), the term is still \( a_{100} = 3 \).

Term calculation here shows that all terms share the same value, demonstrating the structure of a constant sequence.

Arithmetic Sequences

Understanding arithmetic sequences helps distinguish them from constant sequences. An arithmetic sequence features a constant difference between consecutive terms, indicated by adding or subtracting a fixed number from the previous term.
For instance, if an arithmetic sequence starts at 2 with a common difference of 3, the sequence is 2, 5, 8, 11, etc.

• Start with a first term, like 2.
• Add the common difference, such as 3, to find the next term.
• This results in a linear pattern, expanding with each new calculation.

In contrast, a constant sequence remains the same throughout, such as \( a_n = 3 \), where every term is identical without change. While not arithmetic in progression, constant sequences are foundational and underscore the broad possibilities of numeric patterns in sequences.