Problem 11 - Algebra for College Students

Question

Find the 15 th and 30 th terms of the sequence where $$ a_{n}=-5 n-4 . \quad a_{15}=-79 ; a_{30}=-154 $$

Step-by-Step Solution

Answer

The 15th term is -79 and the 30th term is -154.

Step 1: Understand the Sequence Formula

The general formula for the sequence is given by \(a_n = -5n - 4\). This formula will help us calculate any term \(a_n\) by substituting the value of \(n\).

Step 2: Calculate the 15th Term

Substitute \(n = 15\) into the formula to find \(a_{15}\): \[a_{15} = -5(15) - 4 = -75 - 4 = -79\]. This confirms that the 15th term is \(-79\).

Step 3: Calculate the 30th Term

Substitute \(n = 30\) into the formula to find \(a_{30}\): \[a_{30} = -5(30) - 4 = -150 - 4 = -154\]. This confirms that the 30th term is \(-154\).

Step 4: Check Provided Values for Consistency

Verify that the calculated terms match the given values: \(a_{15} = -79\) and \(a_{30} = -154\). Both are consistent with our calculations based on the formula.

Key Concepts

Understanding the Sequence Formula

In the study of algebraic sequences, a sequence formula is a crucial tool that allows us to find any term in a sequence. For this particular sequence, we have the formula \(a_n = -5n - 4\). This formula is a linear expression where \(n\) represents the position of a term in the sequence, such as the 1st, 2nd, or 15th term. The sequence formula is designed to give us a complete description of how each term is related to its position, \(n\).
This can be very helpful because, once the sequence rule is known, any term can be calculated quickly by substituting the corresponding value of \(n\).
Remember, in sequences expressed in this way, the term with \(n\) has a direct dependency on the value of \(n\). This determines how the sequence progresses.

Performing Term Calculation

Term calculation involves using the sequence formula to find specific terms within a sequence, like the 15th or 30th term. To calculate a term, you replace \(n\) in the formula with the term number you are interested in.

• For the 15th term, substitute \(n = 15\) into the formula:
  \[a_{15} = -5(15) - 4 = -75 - 4 = -79\]
• For the 30th term, substitute \(n = 30\) into the formula:
  \[a_{30} = -5(30) - 4 = -150 - 4 = -154\]

These calculations confirm the specific terms of the sequence using basic arithmetic operations. Checking your work ensures that each calculation aligns with expected outcomes, reinforcing the accuracy of the formula and your process.

Substitution in Sequences

Substitution in sequences is the act of plugging specific numbers into the sequence formula to find particular terms. This process is critical in sequence analysis as it establishes how each position translates into a specific term value.
When substituting, be sure to insert the correct \(n\) value to match the intended term. Careful substitution helps confirm that the sequence formula is being applied consistently and accurately across all terms you calculate.
Through substitution, each number plugged into \(n\) will help illustrate the pattern or progression within the sequence, thus validating the properties and rules defined by the sequence, such as our formula \(a_n = -5n - 4\).
Remember, consistent substitution results are a great way to verify the correctness of a sequence rule.