Problem 56 - College Algebra with Corequisite Support

Question

At which term does the sequence \(\left\\{\frac{1}{2187}, \frac{1}{729}, \frac{1}{243}, \frac{1}{81} \quad \ldots\right\\}\) begin to have integer values?

Step-by-Step Solution

Answer

The sequence has integer values starting at the 7th term.

Step 1: Identify the Pattern

The given sequence is \( \left\{ \frac{1}{2187}, \frac{1}{729}, \frac{1}{243}, \frac{1}{81}, \ldots \right\} \). Each term appears to have a denominator that is a power of 3.

Step 2: Express in General Form

Express the \(n\)-th term of the sequence in exponential form: \( a_n = \frac{1}{3^{7-n}} \). This matches the denominators of the terms in the sequence.

Step 3: Find Integer Value Condition

For \( a_n \) to be an integer, the denominator \( 3^{7-n} \) must be equal to \(1\). This occurs when \( 7-n = 0 \), giving us \( n = 7 \).

Step 4: Verify the Solution

Substitute \( n = 7 \) back into the formula \( a_n = \frac{1}{3^{7-n}} \). We find \( a_7 = \frac{1}{3^{7-7}} = \frac{1}{3^0} = 1 \), confirming that \(a_n\) is indeed an integer.

Key Concepts

Integer Values in Sequences

Sequences in algebra often include terms that are fractions, but there may be a point where these terms change to integer values. To determine when this happens, we look for a specific condition.

In the given sequence, the goal is to discover which term results in an integer value. This is done by setting the expression of terms equal to a value where fractions disappear. In other words, the denominator must simplify to 1.

Calculating integer values in sequences involves algebraic manipulation. Check where the form of the sequence results in a whole number. In this problem, calculating the term number for an integer simplifies the sequence's nature and the pattern's exploration.

Exponential Form

In this sequence, each term follows an exponential pattern. This relates to powers of a number, often used in algebra to express repetitive multiplication efficiently.

Here, the sequence's denominators are powers of 3. By expressing each term in exponential form, it's clear that each denominator is expressed as \(3^{7-n}\). This form helps reveal the underlying structure of the sequence and simplifies identifying transformations and patterns.

Exponential notation is crucial for sequences because it offers a concise way to represent a wide range of numbers. It also helps in determining the condition under which terms can be manipulated for further algebraic operations, such as achieving an integer value.

Patterns in Sequences

Recognizing patterns is a key skill in algebra. It allows you to predict the behavior of sequences and anticipate future terms easily.

In the given sequence:

• The denominators follow a clear pattern of being successive powers of 3: 2187, 729, 243, and 81.
• Each term's denominator is a decreasing power of 3, indicating the progression structure.

By understanding these patterns, you can articulate a general formula, predict subsequent terms, or determine when specific conditions occur, such as integer values. Identifying and understanding these patterns make solving problems more intuitive and less algorithmic.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They form the backbone of algebra, providing a way to express sequences.

In this case, the sequence's terms are described by the algebraic expression \(a_n = \frac{1}{3^{7-n}}\). This expression is derived from recognizing the pattern in terms' denominators and using algebraic techniques to express them in a standard form.

Using algebraic expressions helps in computation and ensures that the formula works for any term, not just the ones you initially identify. By manipulating these expressions, you can find specific terms in a sequence, solve equations, or determine other properties. This understanding is essential for both solving math problems and applying mathematical concepts to real-world situations.