Problem 21 - 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. $$ 1,4,7,10, \dots $$

Step-by-Step Solution

Answer

The algebraic expression is \(a_n = 3n - 2\), and the ninth term is 25.

Step 1: Identify the Pattern

Observe that the sequence is 1, 4, 7, 10, ... This sequence increases by 3 each time, so it is an arithmetic sequence with the common difference of 3.

Step 2: General Formula for the Sequence

In an arithmetic sequence, the nth term can be represented as \(a_n = a_1 + (n-1) \times d\), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 1\) and \(d = 3\).

Step 3: Derive the Expression for the Sequence

Substitute \(a_1 = 1\) and \(d = 3\) into the formula: \[ a_n = 1 + (n-1) \times 3 \] Simplify the expression: \[ a_n = 1 + 3n - 3 = 3n - 2 \] So, the algebraic expression for the nth term is \(a_n = 3n - 2\).

Step 4: Find the Ninth Term

To find the ninth term, substitute \(n = 9\) into the expression \(a_n = 3n - 2\): \[ a_9 = 3(9) - 2 \] Calculate: \[ a_9 = 27 - 2 = 25 \] So the ninth term is 25.

Key Concepts

Understanding the Common Difference in Arithmetic Sequences

In an arithmetic sequence, the common difference is the difference between any two consecutive terms. Think of it as the consistent amount by which the sequence increases or decreases from one term to the next. For our sequence, which starts as 1, 4, 7, 10, ..., you can easily spot that each term is increasing by 3. This makes 3 our common difference.

• The common difference can be positive, leading to an increasing sequence.
• It can also be negative, resulting in a decreasing sequence.
• If the common difference is zero, all terms of the sequence are the same.

Recognizing the common difference is crucial as it helps us form the algebraic expression of the sequence. In this example, understanding that the pattern involves adding 3 helps us predict future terms with ease.

Forming an Algebraic Expression for Arithmetic Sequences

Creating an algebraic expression for an arithmetic sequence involves using its common difference and the first term. The algebraic expression is a handy tool, allowing us to find any term in the sequence without listing every previous term.
You use the general formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1) imes d \] Where:

• \( a_n \) is the nth term we want to calculate.
• \( a_1 \) is the first term, which in this sequence is 1.
• \( d \) is the common difference, which for our sequence is 3.

When we substitute the known values, we derive the expression: \[ a_n = 1 + (n-1) imes 3 \] This simplifies to the algebraic expression \( a_n = 3n - 2 \). Having this expression is like having a shortcut to finding any term in the sequence.

Utilizing the Nth Term Formula to Find Sequence Terms

Using the derived algebraic expression, we can easily find the nth term of a sequence without having to manually count each term until we reach n. This is especially useful for arithmetic sequences spanning many terms.For example, to find the ninth term of our sequence, we plug \( n = 9 \) into our expression \( a_n = 3n - 2 \):\[ a_9 = 3(9) - 2 \] Calculating gives us:\[ 27 - 2 = 25 \] So, the ninth term of the sequence is 25.
By using the nth term formula, you can efficiently determine any specific term's value, which is an excellent way to understand how the sequence develops and predict its future values. Knowing how to apply this formula is a key skill in working with arithmetic sequences.