Problem 20 - Algebra 2 and Trigonometry

Question

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 3,6,9,12, \dots $$

Step-by-Step Solution

Answer

The algebraic expression is \(a_n = 3n\) and the ninth term is 27.

Step 1: Identify the type of sequence

The given sequence is 3, 6, 9, 12, ... . Notice that the difference between consecutive terms is constant. For example, \(6 - 3 = 3\), \(9 - 6 = 3\), and \(12 - 9 = 3\). Therefore, this sequence is an arithmetic sequence with a common difference \(d = 3\).

Step 2: Determine the first term

The first term of the sequence, denoted as \(a_1\), is the first number in the sequence. For our sequence, \(a_1 = 3\).

Step 3: Write the algebraic expression for the general term

The general term for an arithmetic sequence is given by the formula \(a_n = a_1 + (n - 1) \, d\), where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. Substituting the values, we have \(a_n = 3 + (n - 1) \, 3\).

Step 4: Simplify the algebraic expression

Simplify the expression obtained in the previous step: \(a_n = 3 + 3(n - 1)\). Distribute the 3: \(a_n = 3 + 3n - 3\). Simplify this to obtain \(a_n = 3n\).

Step 5: Find the ninth term

To find the ninth term of the sequence, substitute \(n = 9\) into the expression for \(a_n\): \(a_9 = 3(9)\). Calculate the value to get \(a_9 = 27\).

Key Concepts

Algebraic Expression for Arithmetic Sequences

An algebraic expression represents a mathematical "recipe" for generating terms of a sequence. For arithmetic sequences, this expression follows a consistent pattern, enabling easy calculation of any term in the sequence. The general form used for arithmetic sequences is - \( a_n = a_1 + (n - 1) \, d \), where: - \( a_1 \) is the first term - \( n \) is the position of the term in the sequence - \( d \) is the common difference between consecutive terms.
In the sequence 3, 6, 9, 12, ..., the first term \( a_1 \) is 3, and the sequence progresses by adding a common difference each time, leading to the formula \( a_n = 3 + (n-1) \,3 \). Simplifying, you combine like terms, resulting in \( a_n = 3n \). Thus, to find any term, simply multiply the term number by 3.

Understanding the Common Difference

The common difference is the heart of an arithmetic sequence. It dictates how each term grows from the previous one.
For example, in the sequence 3, 6, 9, 12, ..., the common difference is 3. It's calculated by subtracting any term from the term that follows it, such as: - \( 6 - 3 = 3 \) - \( 9 - 6 = 3 \) - \( 12 - 9 = 3 \)
This consistent positive interval confirms the sequence's arithmetic nature. Recognizing the common difference not only helps in creating the algebraic expression but also allows quick predictions of future terms in the sequence.

Term of a Sequence

A term in a sequence is an individual element of that sequence, with its position marked by \( n \).
The term's value is often denoted and found using a formula derived from the sequence's characteristics, such as the algebraic expression for arithmetic sequences. In our example, - The ninth term is represented as \( a_9 \).
By substituting \( n = 9 \) into the expression \( a_n = 3n \), the calculation becomes: - \( a_9 = 3 \times 9 = 27 \).
Whether you need the first, ninth, or hundredth term, this substitution and calculation method grant you fast and accurate results.