Problem 57 - College Algebra

Question

For which term does the gcometric sequence \(a_{n}=-36\left(\frac{2}{3}\right)^{n-1}\) first have a non-integer value?

Step-by-Step Solution

Answer

The fourth term, \(a_4\), is the first non-integer value.

Step 1: Understand the Problem

The given sequence is a geometric sequence with the nth term given by \(a_n = -36\left(\frac{2}{3}\right)^{n-1}\). Our task is to find the smallest integer \(n\) for which \(a_n\) is not an integer.

Step 2: Identify Integer Terms

For \(a_n\) to be an integer, \(-36\left(\frac{2}{3}\right)^{n-1}\) must be an integer. The fraction \(\left(\frac{2}{3}\right)^{n-1}\) introduces powers of 2 in the numerator and powers of 3 in the denominator. We need to determine when this fraction causes \(a_n\) to become a non-integer.

Step 3: Check Integer Requirement

Express \(\left(\frac{2}{3}\right)^{n-1}\) as \((2^{n-1})/(3^{n-1})\). For \(a_n\) to be an integer, \(3^{n-1}\) must be a factor of \(-36\). Since \(-36 = -1 \times 2^2 \times 3^2\), \(3^{n-1}\) should divide \(3^2\). Here, \(n-1\) can be 0, 1, or 2.

Step 4: Determine the Smallest Non-Integer Term

The values of \(n-1\) that keep \(a_n\) as an integer are 0, 1, and 2 (as checked from the limits of divisibility by 3 in step 3). Therefore, for \(n = 4\) (since \(n = 3 + 1 = 4\)), \(n-1 = 3\), and \(3^3\) is not a factor of \(-36\). So, \(a_4\) is not an integer.

Key Concepts

Understanding Integer Sequences

An integer sequence consists of numbers that are all integers. These numbers follow a particular pattern or rule. In mathematics, integer sequences are often very predictable. Common examples include the sequence of whole numbers or even numbers.

When dealing with integer sequences, it is crucial to identify the rule governing the sequence. For example, the nth term of a sequence might be defined by a formula like \( a_n = -36\left(\frac{2}{3}\right)^{n-1} \) as mentioned in our exercise.

• Integer: A number without a decimal or fractional part.
• Term: Each individual element or number in a sequence.
• Sequence: An ordered list of numbers defined by a specific rule.

Understanding when a sequence is made of integers only involves checking if each term remains a whole number under the given formula.

Exploring Non-Integer Values

Non-integer values occur in a sequence when a term breaks away from being a whole number. Such values often appear as fractions or decimals, indicating they are not complete integers.

In our specific sequence \(a_n = -36\left(\frac{2}{3}\right)^{n-1}\), observing when \(a_n\) becomes a non-integer helps us understand the divisor rule. Here, the term becomes non-integer when the fraction \(\left(\frac{2}{3}\right)^{n-1}\) introduces a part that cannot divide the integer part of the formula evenly.

Key points include:

• Fraction: A number represented as \(\frac{numerator}{denominator}\).
• Non-integer: A number that isn't a whole or complete integer.
• Understanding when non-integers appear can significantly affect calculations and progression in mathematics.

By determining the smallest \(n\) where a non-integer appears, one can better analyze the properties of such sequences.

Geometric Progression Unveiled

A geometric progression (or geometric sequence) is a sequence of numbers where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

In the provided sequence \( a_n = -36\left(\frac{2}{3}\right)^{n-1} \), the common ratio is \(\frac{2}{3}\). This means each term is obtained by multiplying the previous term by \(\frac{2}{3}\).

Characteristics of geometric sequences include:

• Common ratio: \(\frac{2}{3}\) in this scenario, which dictates how each term in the sequence is calculated from the previous term.
• Exponential growth or decay depending on if the common ratio is greater than or less than 1.
• The nth term formula: Defines any term in a geometric sequence, such as \(a_n = -36\left(\frac{2}{3}\right)^{n-1}\).

Understanding geometric progressions helps in predicting the behavior of a sequence and the effects of applying a ratio repeatedly across terms.

Factorization in Algebraic Expressions

Factorization is the process of breaking down an expression into a product of its factors. It is a critical skill in algebra as it simplifies expressions and solves equations. In the context of our geometric sequence exercise, factorization helps explain why certain terms are integers and others are not.

For \( a_n \) to remain an integer, every term in the denominator of the fraction must be a factor of the leading constant (here, \(-36\)). When this condition is not met, it results in a non-integer term.

Here's what you should remember about factorization:

• Expression: Mathematical sentence involving numbers, variables, and operation symbols.
• Factor: An element 'multiplied' with another to get an expression, like \( -36 = -1 \times 2^2 \times 3^2 \).
• Factorization can simplify complex calculations and uncover the structure and characteristics within algebraic expressions.

This concept is particularly useful in identifying when sequences deviate from being entirely made of integer terms, as seen in this geometric sequence exercise.