Problem 12 - Algebra and Trigonometry

Question

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=3 $$

Step-by-Step Solution

Answer

All terms are 3, including the 100th.

Step 1: Identify the sequence type

Observe the given formula for the sequence, \[a_n = 3\] This formula implies that the sequence is a constant sequence where each term is the same.

Step 2: Write the first four terms

Since each term in the sequence is the constant value 3, the first four terms are simply 3, 3, 3, and 3.

Step 3: Find the 100th term

Because the sequence is constant, every term in the sequence, including the 100th term, is the same. Therefore, the 100th term is:\[a_{100} = 3\]

Key Concepts

Sequence Types

In mathematics, sequences are an ordered list of numbers that follow a specific pattern or rule. There are several types of sequences, such as arithmetic, geometric, and constant sequences. Here, we're focusing on constant sequences, which is a straightforward type of sequence.
A constant sequence is one where every term is the same. It does not change regardless of which term you are looking at. This type occurs when a sequence is defined by a constant mathematical formula, such as \( a_n = 3 \), meaning that every term \( a_n \) in the sequence is equal to 3.
Understanding sequence types is crucial as they lay the foundation for identifying patterns and solving equations related to sequences in mathematics.

Sequence Terms

The terms of a sequence are the individual elements or numbers that make up the sequence. In a constant sequence, each of these terms is identical. For example, if the sequence is defined by \( a_n = 3 \), the sequence terms are all 3s, regardless of their position in the sequence.
Here is how it looks for the first four terms:

• First term: 3
• Second term: 3
• Third term: 3
• Fourth term: 3

It's essential to recognize that a sequence like this offers the simplest representation, as each term is predictable and defined by the sequence formula.

Mathematical Formula

A mathematical formula for a sequence provides a rule, allowing us to calculate any term in the sequence. For constant sequences, the formula is particularly simple. The example \( a_n = 3 \) signifies that no matter which term \( n \) you are calculating, it will always equate to 3.
The formula itself is the key to understanding and computing sequences:

• The structure \( a_n \) represents the \( n \)-th term.
• The equals sign and the number (3 in this case) denote that each term is equal to this constant value.

A strong grasp of these formulas facilitates solving problems and understanding more complex sequence types.

Nth Term of a Sequence

The \( n \)-th term of a sequence corresponds to the term at the \( n \)-th position in the sequence. To find any specific term, like the 100th term, you apply the sequence's mathematical formula.
For a constant sequence where \( a_n = 3 \), \( a_{100} \) is simply calculated as 3. There's no complexity or change in values, which makes these sequences unique and easy to handle. They exemplify the idea that one can determine any sequence term using its defining formula, ensuring predictability and simplicity in calculations.