Problem 12 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The first four terms are 3, 3, 3, 3, and the 100th term is 3.

Step 1: Identify the Type of Sequence

The given sequence is represented by the equation \( a_n = 3 \). This means that every term in the sequence is 3. Since each term is the same, this sequence is a constant sequence.

Step 2: Write the General Formula for the Sequence

The general formula given in the problem is \( a_n = 3 \). This means that for any term \( n \) in the sequence, the value will be 3.

Step 3: Calculate the First Four Terms

Using the formula \( a_n = 3 \), calculate the first four terms:- The 1st term: \( a_1 = 3 \)- The 2nd term: \( a_2 = 3 \)- The 3rd term: \( a_3 = 3 \)- The 4th term: \( a_4 = 3 \)Thus, the first four terms are 3, 3, 3, 3.

Step 4: Find the 100th Term

Using the same formula \( a_n = 3 \), we find:- The 100th term: \( a_{100} = 3 \)The 100th term is also 3.

Key Concepts

Sequences in Mathematics

In mathematics, a sequence is a set of numbers arranged in a specific order following a defined rule. These numbers, called terms, can be infinite or finite in number. Each term is typically generated according to a rule or function, often depending on its position within the sequence, denoted by an index like \( n \).

Sequences occur everywhere in mathematics and are very useful for modeling real-world phenomena, such as populations, sound waves, or even movement patterns. Understanding sequences helps us explore how things change and develop systematically over time.

Mathematically, sequences can be categorized into various types including arithmetic, geometric, and constant sequences. Each type has specific properties and formulas for calculating its terms.

• An arithmetic sequence has a constant difference between its consecutive terms.


• A geometric sequence has a constant ratio between its consecutive terms.


• A constant sequence is where every term is identical.

Understanding these categories helps in identifying the pattern and computing various terms effectively.

Arithmetic Sequences

An arithmetic sequence is a type of sequence where each term is formed by adding a fixed, constant value, known as the common difference, to the previous term.

For an arithmetic sequence with a first term \( a_1 \) and a common difference \( d \), the \( n \)-th term can be calculated using the formula:
\[ a_n = a_1 + (n-1) \, d \]

This formula helps in determining any term within the sequence without the need to calculate all preceding terms. For example, if the first term is 2 and the common difference is 3, the sequence would be 2, 5, 8, 11, and so on.

• It is important to note that if the common difference \( d \) is positive, the sequence will increase.


• If \( d \) is negative, the sequence will decrease.


• If \( d \) is zero, it becomes a constant sequence.

Arithmetic sequences are prevalent in real-life situations where things change at a constant rate, such as savings growth over time at a constant interest rate.

General Term of a Sequence

The general term of a sequence is a formula that helps us find any term of the sequence based on its position. Knowing the general term is essential because it allows us to compute terms without having to manually construct the entire list.

For constant sequences, like the one given by \( a_n = 3 \), the general term is very straightforward because each term is the same, no matter the position \( n \). This makes it easy to determine terms, even for a large index, such as the 100th term, since you know all terms equal 3.

• For other types of sequences, such as arithmetic or geometric, the general term will involve more complex calculations depending on their unique formulas.


• Understanding how to derive the general term can simplify complex sequence analysis and make understanding patterns much easier.

Recognizing the general term formula is a fundamental part of working with sequences, easing the computation of terms, and revealing the sequence's behaviour.