Problem 1 - Algebra for College Students

Question

Find the 15th term of the sequence for which \(a_{n}=-n^{2}-1 . \quad-226\)

Step-by-Step Solution

Answer

The 15th term of the sequence is -226.

Step 1: Understand the Sequence Formula

The formula given for the sequence is \(a_n = -n^2 - 1\). This formula defines how each term in the sequence \(a_n\) relates to its position, \(n\), in the sequence.

Step 2: Identify the Term's Position

We need to find the 15th term of the sequence. This means we are looking for \(a_{15}\), where \(n = 15\).

Step 3: Substitute the Position into the Formula

Substitute \(n = 15\) into the sequence formula to find \(a_{15}\).\[a_{15} = -(15)^2 - 1\]

Step 4: Calculate the Answer

Evaluate \((15)^2\) which is 225.Then calculate \(-225 - 1\) which equals \(-226\).Thus, \(a_{15} = -226\).

Key Concepts

Sequence Formula in Arithmetic Sequences

An arithmetic sequence is a number pattern defined by a specific rule or formula. The formula gives us a mathematical expression to find the terms based on their position. For instance, the sequence formula provided, \(a_n = -n^2 - 1\), specifies how each number in the sequence relates to its order. In this formula, \(a_n\) represents the term in the sequence at position \(n\). The sequence formula usually involves arithmetic operations like addition, subtraction, or combinations with exponents and coefficients. This helps in predicting the terms without listing all previous numbers. In any sequence, understanding and applying the formula correctly is essential for identifying terms efficiently.

Understanding Term Position in Sequences

In sequences, the position of a term, often denoted as \(n\), is an integral part of identifying the specific term we want in the sequence. Each term corresponds to a unique position number. For example, term 1 is at position 1, term 2 is position 2, and so on. Understanding the term position is crucial when using the sequence formula, as it tells us which term to calculate. In the exercise, we are interested in finding the 15th term of the sequence. Thus, the term position here is \(n = 15\). It's vital to substitute this exact position into the sequence formula to find the correct term value.

Applying the Substitution Method

The substitution method is a straightforward technique used in mathematics to find specific values in formulas. When working with sequences, once a term position is identified, you substitute this number into the sequence formula to determine the value of that sequence term.
For example, if we want to find \(a_{15}\) in the sequence defined by \(a_n = -n^2 - 1\), we substitute \(n = 15\) into the formula like so:

• Calculate \((15)^2\), which results in 225.
• Then, substitute \(n\) and simplify: \(a_{15} = -(225) - 1\).
• Perform the arithmetic operation to get the final answer: \(-226\).

This method proves efficient in quickly finding terms of the sequence without manually extending it. Understanding substitution will make using any sequence formulas much more manageable.