Problem 56 - Algebra 2

Question

Find the indicated term of each arithmetic sequence. \(a_{1}=5, d=\frac{1}{3}, n=12\)

Step-by-Step Solution

Answer

The 12th term is \(\frac{26}{3}\).

Step 1: Understand the Problem

We need to find the 12th term of an arithmetic sequence given the first term, \(a_1 = 5\), and the common difference, \(d = \frac{1}{3}\). The formula to find the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\).

Step 2: Substitute Known Values

Substitute the known values \(a_1 = 5\), \(d = \frac{1}{3}\), and \(n = 12\) into the formula: \[a_n = 5 + (12-1) \cdot \frac{1}{3}\]

Step 3: Simplify Inside the Parentheses

Calculate \(12-1\) which is \(11\). Substitute back into the equation: \[a_n = 5 + 11 \cdot \frac{1}{3}\]

Step 4: Multiply to Find the Term Addition

Multiply the common difference by the number of terms minus one: \[11 \cdot \frac{1}{3} = \frac{11}{3}\]

Step 5: Add to Find the N-th Term

Add the result from the multiplication to the first term of the sequence: \[a_n = 5 + \frac{11}{3}\]

Step 6: Convert to Common Denominator

Convert \(5\) to a fraction with the same denominator as \(\frac{11}{3}\). \[5 = \frac{15}{3}\]

Step 7: Perform the Addition

Add \(\frac{15}{3}\) and \(\frac{11}{3}\): \[a_n = \frac{15}{3} + \frac{11}{3} = \frac{26}{3}\]

Step 8: Simplify the Fraction If Necessary

The fraction \(\frac{26}{3}\) cannot be simplified further, so the 12th term, \(a_{12}\), is \(\frac{26}{3}\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the common difference, denoted as \(d\), is key to understanding how the sequence progresses. The common difference is the amount added, or subtracted if negative, to each term to yield the following term. This difference remains constant throughout the sequence.
For example, in the given sequence with first term \(a_1 = 5\) and common difference \(d = \frac{1}{3}\), each term is found by adding \(\frac{1}{3}\) to the previous term. This characteristic is what defines the sequence as "arithmetic."

• The common difference helps determine how quickly the sequence increases or decreases.
• It is derived from the difference between any two consecutive terms. In formula terms: \(d = a_{n+1} - a_n\).

Understanding the common difference is crucial, as it is directly used in finding any term within the sequence.

Exploring the N-th Term Formula

The n-th term formula is a powerful tool in arithmetic sequences, allowing us to find any term in the sequence without listing all preceding terms. The formula is given by:
\[a_n = a_1 + (n-1)\cdot d\]
Here, \(a_n\) represents the n-th term in the sequence, \(a_1\) is the first term, and \(d\) is the common difference. The value \(n\) is the position of the term in the sequence.

• This formula eliminates the need to manually calculate each preceding term.
• It relies heavily on the value of the common difference as it determines the step to reach the desired term from the first term.

Using this formula is straightforward and immensely beneficial, especially for large values of \(n\).

Performing Term Calculation

Calculating a specific term in an arithmetic sequence using the n-th term formula involves straightforward substitution and arithmetic. Here's how to approach it:
With the given values: \(a_1 = 5\), \(d = \frac{1}{3}\), and \(n = 12\), substitute these into the n-th term formula:
\[a_{12} = 5 + (12-1) \cdot \frac{1}{3}\]
First calculate the expression inside the parentheses: \(12-1 = 11\). Followed by multiplying it with the common difference:
\[11 \cdot \frac{1}{3} = \frac{11}{3}\]
Finally, add this result to the first term:
\[a_{12} = 5 + \frac{11}{3}\].
This approach bridges finding specific terms directly.

Understanding Sequence Simplification

When calculating terms in arithmetic sequences, results often require simplification, especially when they involve fractions. In our example, the task involves adding \(5\) and \(\frac{11}{3}\):

• First, convert \(5\) into an equivalent fraction with a denominator of \(3\):
  \(5 = \frac{15}{3}\).
• Then, add the fractions: \(\frac{15}{3} + \frac{11}{3} = \frac{26}{3}\).

The last step is checking if the fraction can be simplified further. In this case, \(\frac{26}{3}\) is already in its simplest form. Sequence simplification ensures that the terms are expressed in the simplest possible values, which makes them easier to interpret.