Problem 2 - College Algebra with Modeling and Visualization

Question

Find the first four terms of the sequence. \(a_{n}=3(n-1)+5\)

Step-by-Step Solution

Answer

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

Step 1: Write Down the Sequence Formula

The sequence is defined by the formula \( a_n = 3(n-1) + 5 \). This formula expresses how to find the \( n \)-th term of the sequence.

Step 2: Calculate the First Term

To find the first term, substitute \( n = 1 \) into the formula: \( a_1 = 3(1-1) + 5 = 3 \times 0 + 5 = 5 \). Thus, the first term is 5.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula to find the second term: \( a_2 = 3(2-1) + 5 = 3 \times 1 + 5 = 8 \). Thus, the second term is 8.

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula to find the third term: \( a_3 = 3(3-1) + 5 = 3 \times 2 + 5 = 11 \). Thus, the third term is 11.

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula to find the fourth term: \( a_4 = 3(4-1) + 5 = 3 \times 3 + 5 = 14 \). Thus, the fourth term is 14.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a simple yet fascinating concept in algebra. It's a sequence where the difference between consecutive terms is always the same. This fixed gap is called the "common difference." Understanding this idea helps in identifying the pattern within the sequence.
For example, consider the sequence 5, 8, 11, 14. Here, each term increases by 3 (the common difference). Such sequences are straightforward to work with because you can easily predict future terms by consistently adding the common difference.

• Start with any number as the first term.
• Add the common difference to find the next number.
• Repeat the process to extend the sequence.

This clear pattern makes arithmetic sequences easy to handle and highly predictable.

Sequence Formula

The sequence formula is a powerful tool for establishing any term in a sequence without having to count each one-by-one. You plug in the position of the term you want into the formula, and voilà!
In the given problem, the formula for the sequence is \( a_n = 3(n-1) + 5 \). This formula tells you everything you need to know:

• \( n \) is the position or index of the term.
• The formula incorporates a pattern leveraging the common difference.

For example, to find the first term, set \( n = 1 \), and for the second term, use \( n = 2 \). The formula becomes your reliable shortcut to finding any term in the sequence efficiently.

Finding Terms in a Sequence

Finding specific terms in a sequence is quite exciting when you apply the known formula because it immediately shows the precision and efficiency in mathematics.
Let's break it down. To find each term, substitute the term number \( n \) into the formula. So for the sequence given by \( a_n = 3(n-1) + 5 \):

• For the first term \((n=1)\), find \( a_1 = 3(1-1) + 5 = 5 \).
• For the second term \((n=2)\), calculate \( a_2 = 3(2-1) + 5 = 8 \).
• For the third term \((n=3)\), determine \( a_3 = 3(3-1) + 5 = 11 \).
• For the fourth term \((n=4)\), evaluate \( a_4 = 3(4-1) + 5 = 14 \).

Each calculation involves simple substitution and arithmetic. This procedure not only confirms the consistency of the sequence but also affirms your understanding of how sequences work.