Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=12-3 n $$

Step-by-Step Solution

Answer

The first five terms are 9, 6, 3, 0, -3.

Step 1: Identify the sequence formula

The sequence is given by the formula \( a_{n} = 12 - 3n \). This formula expresses each term of the sequence in terms of \( n \), which represents the position of the term in the sequence.

Step 2: Find the first term of the sequence

Substitute \( n = 1 \) into the sequence formula:\[ a_{1} = 12 - 3(1) = 12 - 3 = 9 \].The first term of the sequence is 9.

Step 3: Calculate the second term of the sequence

Substitute \( n = 2 \) into the sequence formula:\[ a_{2} = 12 - 3(2) = 12 - 6 = 6 \].The second term of the sequence is 6.

Step 4: Determine the third term of the sequence

Substitute \( n = 3 \) into the sequence formula:\[ a_{3} = 12 - 3(3) = 12 - 9 = 3 \].The third term of the sequence is 3.

Step 5: Evaluate the fourth term of the sequence

Substitute \( n = 4 \) into the sequence formula:\[ a_{4} = 12 - 3(4) = 12 - 12 = 0 \].The fourth term of the sequence is 0.

Step 6: Compute the fifth term of the sequence

Substitute \( n = 5 \) into the sequence formula:\[ a_{5} = 12 - 3(5) = 12 - 15 = -3 \].The fifth term of the sequence is -3.

Key Concepts

Sequence Formula

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. To describe this sequence, we use what is known as the "sequence formula."
The sequence formula helps us express any term in the sequence as a function of its position. Here, our formula is given by \( a_{n} = 12 - 3n \).

• \( a_{n} \) denotes the nth term of the sequence.
• 12 is known as the initial term adjustment and can be considered a starting point.
• -3 represents the difference between consecutive terms, also referred to as the common difference.
• \( n \) is the position number of the term within the sequence.

Understanding this formula will enable you to find any term in the sequence without having to list all previous terms.

Term Calculation

Calculating a specific term in an arithmetic sequence becomes straightforward when you use the sequence formula. The term calculation involves substituting the position number \( n \) into the sequence formula and then simplifying to find the term.
For example, to calculate the first term, you substitute \( n = 1 \) into the formula:\[ a_{1} = 12 - 3(1) = 12 - 3 = 9. \]The same principle applies to find the second, third, and subsequent terms. Simply replace \( n \) in the formula with the desired position number.

• Second term: \( n = 2 \), resulting in \( a_{2} = 6. \)
• Third term: \( n = 3 \), resulting in \( a_{3} = 3. \)
• Fourth term: \( n = 4 \), resulting in \( a_{4} = 0. \)
• Fifth term: \( n = 5 \), resulting in \( a_{5} = -3. \)

Each step involves simple arithmetic, ensuring you arrive at the desired term effortlessly.

Position in Sequence

Understanding the position in a sequence is essential for finding the correct term using the sequence formula. The position, often denoted as \( n \), indicates where a term appears in the sequence.

• The first term is at position \( n = 1 \), the second at \( n = 2 \), and so on.
• This "positional index" is crucial as it guides how we apply the sequence formula to calculate the terms accurately.

By identifying the position of a term, you can substitute it into the sequence formula to find the numerical value of that term.The sequence's positions are sequential integers, making it easy to apply the formula accurately without errors.
To shift from theoretical to practical application, always ensure you identify your desired position correctly before substituting it into the sequence formula. This method ensures all calculations are precise, offering you an efficient way to navigate through arithmetic sequences.