Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=\frac{n}{2} $$

Step-by-Step Solution

Answer

The first five terms are 0.5, 1, 1.5, 2, and 2.5.

Step 1: Understand the sequence formula

The sequence given is defined by the formula \( a_n = \frac{n}{2} \). This means that the value of each term in the sequence is obtained by dividing \( n \), the position of the term, by 2.

Step 2: Compute the first term

To find the first term \( a_1 \), substitute \( n = 1 \) into the formula:
\[ a_1 = \frac{1}{2} = 0.5 \]

Step 3: Compute the second term

To find the second term \( a_2 \), substitute \( n = 2 \) into the formula:
\[ a_2 = \frac{2}{2} = 1 \]

Step 4: Compute the third term

To find the third term \( a_3 \), substitute \( n = 3 \) into the formula:
\[ a_3 = \frac{3}{2} = 1.5 \]

Step 5: Compute the fourth term

To find the fourth term \( a_4 \), substitute \( n = 4 \) into the formula:
\[ a_4 = \frac{4}{2} = 2 \]

Step 6: Compute the fifth term

To find the fifth term \( a_5 \), substitute \( n = 5 \) into the formula:
\[ a_5 = \frac{5}{2} = 2.5 \]

Key Concepts

Sequence Formula

An arithmetic sequence is defined by a sequence formula. This formula provides a way to find the value of each term based on its position. For instance, when given a sequence formula like \( a_n = \frac{n}{2} \), it means the term's value is calculated by dividing the position number \( n \) by 2.
The sequence formula is essential because it gives us a rule or pattern for the entire sequence. Without this, we'd be guessing the value of each term!
The formula allows us to systematically calculate any term in the sequence as long as we know its position. Simply plug in the position number into the formula and perform the arithmetic operation, and you'll get your desired term.

Term Calculation

Term calculation using the sequence formula involves substituting the position number into the given formula. Once you know the sequence formula, calculating individual terms becomes straightforward.
Let's look at an example with \( a_n = \frac{n}{2} \):

• To find the first term, substitute \( n = 1 \) into the formula: \( a_1 = \frac{1}{2} = 0.5 \).
• For the second term, use \( n = 2 \): \( a_2 = \frac{2}{2} = 1 \).
• Similarly, the third term is \( a_3 = \frac{3}{2} = 1.5 \).

By continuing this pattern, you can calculate as many terms as needed from the sequence. Remember, each value gives you a clear understanding of how different terms relate as they grow in arithmetic sequence.

Position of Term

Understanding how the position of a term connects to its value is a key concept in sequences. The position of the term is simply the order number in the sequence. For example, the first term is in position 1, the second in position 2, and so on. Each position number, \( n \), corresponds directly to the structured rule provided by the sequence formula.
This relationship is critical because the formula articulates how each position is transformed into a term value. For the formula \( a_n = \frac{n}{2} \), every term's value grows as the term's position increases, since \( n \) in the numerator of the fraction gets larger.
The pattern exhibited in such a sequence helps provide predictability. As you calculate further, knowing the term positions allows students to systematically explore and understand the arithmetic growth exhibited across the sequence.