Problem 2 - Algebra and Trigonometry

Question

write the first four terms of each sequence whose general term is given. $$ a_{n}=4 n-1 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are: 3, 7, 11, 15

Step 1: Determining the First Term

To find the first term of the sequence i.e. \(a_1\), substitute \(n=1\) in the general term \(a_{n}=4n-1\). So, \(a_1 = 4(1) - 1 = 4 - 1 = 3\)

Step 2: Determining the Second Term

To find the second term of the sequence i.e. \(a_2\), substitute \(n=2\) in the general term \(a_{n}=4n-1\). So, \(a_2 = 4(2) - 1= 8 - 1 = 7\)

Step 3: Determining the Third Term

To find the third term of the sequence i.e. \(a_3\), substitute \(n=3\) in the general term \(a_{n}=4n-1\). So, \(a_3 = 4(3) - 1 = 12 - 1 = 11\)

Step 4: Determining the Fourth Term

To find the fourth term of the sequence i.e. \(a_4\), substitute \(n=4\) in the general term \(a_{n}=4n-1\). So, \(a_4 = 4(4) - 1 = 16 - 1 = 15\)

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This difference is referred to as the common difference, and it is quite essential as it defines the arithmetic nature of the sequence.

For example, if we start with the number 2 and keep adding 3 to get the next term, the sequence would be 2, 5, 8, 11, and so on. Here, the common difference is 3. Recognizing the pattern of an arithmetic sequence helps us predict subsequent values and the behaviors of the sequence over time.

General Term of a Sequence

The general term of a sequence, often denoted by the variable term 'a_n', provides a formulaic representation allowing you to find any term in the sequence without listing all the preceding terms. This is particularly useful for arithmetic sequences where each term can be calculated independently of the others.

A general term for an arithmetic sequence can be written as: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Calculating the general term is like having a map, letting you jump to any point in the sequence directly.

Finding Terms of a Sequence

To find specific terms of a sequence, we apply the general term formula by substituting the position number we are interested in for \( n \.\)

For instance, if we wish to find the 100th term of an arithmetic sequence without having to write out the first 99 terms, we can simply plug in 100 for \( n \) in the general term formula. This process saves time and allows for quick calculations of terms in long sequences. Always double-check your substitution to ensure the correct term is calculated.

Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra, consisting of numbers, variables (like \( n \) in sequences), and operations (like addition and multiplication). In the context of sequences, these expressions define the rule or pattern by which the sequence is created.

An algebraic expression for a sequence, like the one in the provided example \( a_n = 4n - 1 \), makes it straightforward to compute terms. The formula itself encapsulates the relationship between the terms of the sequence and their position, which is invaluable when you're dealing with sequences that extend to many terms.