Problem 21 - Basic Engineering Mathematics

Question

Which term of the series \(2187,729,243, \ldots\) is \(\frac{1}{4} ?\)

Step-by-Step Solution

Answer

The term \(\frac{1}{4}\) is not part of the series.

Step 1: Identify the Pattern

First, we identify the pattern of the series. The sequence is: \(2187, 729, 243, \ldots\). Each term appears to be a power of 3: \(2187 = 3^7\), \(729 = 3^6\), \(243 = 3^5\). Therefore, the sequence can be expressed as \(a_n = 3^{8-n}\).

Step 2: Set Up Equation

We know that one of the terms in the series is \(\frac{1}{4}\). We need to find \(n\) such that \(a_n = \frac{1}{4}\). Substituting into the series expression, we get: \(3^{8-n} = \frac{1}{4}\).

Step 3: Solve for n

We need to compare powers of 3 and 4. Rewrite \(\frac{1}{4}\) with a base of 3: \(\frac{1}{4} = 4^{-1}\), and \(4 = (2^2) = (3^{0.6309})^2 = 3^{1.2618}\). Therefore, \(\frac{1}{4} = 3^{-1.2618}\). Thus, we equate: \[ 3^{8-n} = 3^{-1.2618} \]. By equating the powers of 3, we have \(8-n = -1.2618\).

Step 4: Calculate the Term Number

Solve the equation from the previous step, \(8 - n = -1.2618\). Rearrange to find \(n\): \[ n = 8 + 1.2618 = 9.2618 \]. Since \(n\) must be a whole number, and \(n = 9.2618\) is not whole, \(\frac{1}{4}\) is not an exact term in the series.

Key Concepts

Geometric Progression

A geometric progression (or geometric series) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
In the original exercise, the sequence starts with 2187 and each subsequent term is a result of multiplying by a base of 3 raised to a decreasing power.
For example, 2187 can be expressed as the power of 3: \( 2187 = 3^7 \). The sequence then continues with terms like \( 729 = 3^6 \), \( 243 = 3^5 \), and so on, indicating that the common ratio is \( \frac{1}{3} \).

• The formula for the nth term of a geometric progression is \( a_n = ar^{(n-1)} \), where \( a \) is the first term and \( r \) is the common ratio.
• In this sequence: \( a = 2187 \) and \( r = \frac{1}{3} \), and it can be rewritten as \( a_n = 3^{8-n} \).

Understanding this pattern helps in expressing any term of the series in terms of its position, which is crucial for solving progression problems.

Solving for Unknown Terms

When asked to find a specific term in a series that is expressed in a non-standard form, we need to employ some strategies to find the unknown position, \( n \).
In the exercise, we needed to determine which term would equal \( \frac{1}{4} \). To do this, we set up the equation \( a_n = \frac{1}{4} \).
Given the sequence formula \( a_n = 3^{8-n} \), we solve for \( n \) by equating this to the desired term:

• Set \( 3^{8-n} = \frac{1}{4} \).
• Convert \( \frac{1}{4} \) into base 3 terms: \( \frac{1}{4} = 3^{-1.2618} \) through mathematical transformation.
• This results in the equation \( 8-n = -1.2618 \).
• Solve for \( n \) giving \( n = 9.2618 \).

Unfortunately, since \( n \) is not a whole number, \( \frac{1}{4} \) is not an exact term in this series, illustrating an important lesson in recognizing when a term is not naturally part of the sequence.

Pattern Identification

Identifying patterns in sequences plays a crucial role in solving many algebraic problems, notably those involving series or sequences.
In our exercise, pattern identification starts with recognizing that each term in the sequence is a power of 3 with powers that decrease by 1 as you move along the terms.

• Begin by expressing terms in their factorized forms: like \( 2187 = 3^7 \), \( 729 = 3^6 \), etc.
• Recognize the geometric pattern: each term is the previous term divided by 3, illuminating the consistent pattern, or common ratio, of \( \frac{1}{3} \).
• Form the general equation for any term: \( a_n = 3^{8-n} \), suitable for mathematical manipulation such as solving for unknown terms.

Having a keen eye for these patterns can significantly make problem-solving more intuitive and pave the way for precise solutions.