Problem 1 - Algebra and Trigonometry

Question

write the first four terms of each sequence whose general term is given. $$ a_{n}=3 n+2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 5, 8, 11, and 14.

Step 1: Compute the first term

To find the first term, substitute \(n=1\) into \(a_{n}=3n+2\). Thus, \(a_{1}=3*1+2 = 5\).

Step 2: Compute the second term

To find the second term, substitute \(n=2\) into \(a_{n}=3n+2\). Thus, \(a_{2}=3*2+2 = 8\).

Step 3: Compute the third term

To find the third term, substitute \(n=3\) into \(a_{n}=3n+2\). Therefore, \(a_{3}=3*3+2 = 11\).

Step 4: Compute the fourth term

To find the fourth term, substitute \(n=4\) into \(a_{n}=3n+2\). Therefore, \(a_{4}=3*4+2 = 14\).

Key Concepts

General Term Formula

A general term formula is a mathematical expression that defines any term in a sequence as a function of its position. For arithmetic sequences, this formula is crucial to determining the sequence's elements without listing them all. The general term formula in our example is given as \( a_{n} = 3n + 2 \).

• \( a_{n} \) denotes the nth term in the sequence.
• \( n \) represents the position of the term in the sequence.
• The formula \( 3n + 2 \) indicates how each term can be calculated.

This formula helps you find specific terms quickly, by simply substituting the term's position into the equation.

Substitution Method

The substitution method involves replacing variables in equations with specific values to solve for unknowns or compute results. In the context of arithmetic sequences, it's used to find specific terms from the general term formula.

• Choose an integer \( n \) that represents the term position you seek.
• Substitute \( n \) into the general term formula \( a_{n} = 3n + 2 \).
• Solve the equation to find the value of \( a_{n} \).

For example, if you substitute \( n = 1 \), you'll find the first term of the sequence: \( a_{1} = 3 \times 1 + 2 = 5 \). This step-by-step method is reliable for finding any term in a sequence.

Sequence Terms

Sequence terms refer to the individual elements that make up a sequence. Each term has a specific position and value, and they can be generated using the general term formula. In arithmetic sequences, like our example, the terms increase or decrease by the same amount.

• First Term: Find by substituting \( n = 1 \), resulting in \( a_{1} = 5 \).
• Second Term: Find by substituting \( n = 2 \), resulting in \( a_{2} = 8 \).
• Third Term: Find by substituting \( n = 3 \), resulting in \( a_{3} = 11 \).
• Fourth Term: Find by substituting \( n = 4 \), resulting in \( a_{4} = 14 \).

Each term is part of the larger pattern, and understanding individual sequence terms helps to analyze and predict the sequence as a whole.

Pattern Recognition

Pattern recognition in sequences involves identifying how terms change as you progress in the sequence. Recognizing these patterns is key to understanding and predicting sequences beyond merely calculating each term.

• In an arithmetic sequence, each term increases by a constant difference.
• Use the difference to understand the growth or reduction structure of the sequence.

For example, in our sequence, each term increases by 3 from the previous term (e.g., from 5 to 8, from 8 to 11, etc.). Recognizing this pattern confirms that the sequence is arithmetic and follows the general term formula \( a_{n} = 3n + 2 \), providing insight into the nature of the sequence's progression.