Problem 31 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 144,-12,1,-\frac{1}{12}, \dots $$

Step-by-Step Solution

Answer

Common ratio: \(-\frac{1}{12}\), Fifth term: \(\frac{1}{144}\), nth term: \(144 \cdot \left(-\frac{1}{12}\right)^{n-1}\).

Step 1: Identify the Common Ratio

To find the common ratio \( r \) of a geometric sequence, divide any term by the previous term. Here, divide the second term by the first term: \(-12 \div 144 = -\frac{1}{12} \). Thus, the common ratio is \(-\frac{1}{12}\).

Step 2: Find the Fifth Term

The general form of a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term, and \( r \) is the common ratio. For the fifth term \( a_5 \), substitute \( n = 5 \), \( a_1 = 144 \), and \( r = -\frac{1}{12} \):\[a_5 = 144 \cdot \left(-\frac{1}{12}\right)^{5-1} = 144 \cdot \left(-\frac{1}{12}\right)^4 = 144 \cdot \frac{1}{20736}= \frac{144}{20736} = \frac{1}{144}\]

Step 3: Derive the nth Term Formula

Substitute the values found into the general formula of a geometric sequence:\[a_n = a_1 \cdot r^{n-1} = 144 \cdot \left(-\frac{1}{12}\right)^{n-1}.\]This is the formula for the nth term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is crucial for identifying the behavior of the sequence. It's what we multiply each term by to get to the next one.
To find the common ratio (\( r \)), you can simply divide any term by the previous term in the sequence.

• For example, with our sequence: 144, -12, 1, -\( \frac{1}{12} \), the common ratio is found by dividing -12 by 144. This gives us \(-\frac{1}{12}\).
• This constant multiplier determines if the sequence grows or shrinks. Here, because our ratio is negative and less than one, the terms alternate in sign and decrease in absolute value.

Understanding the common ratio helps predict the sequence's progression without calculating every term individually.

Nth Term Formula

The nth term formula of a geometric sequence helps to find any term without listing all the previous ones.
The formula for the nth term is given by:\[a_n = a_1 \cdot r^{n-1}\]where:

• \( a_n \) is the nth term.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term.

By plugging values into this formula, you can easily find any term in the sequence. For instance, the first term (\( a_1 \)) is given as 144, and the common ratio (\( r \)) is \(-\frac{1}{12}\).
So, the formula becomes:\[a_n = 144 \cdot (-\frac{1}{12})^{n-1}.\]This allows computation of any term even for large values of \( n \).

Geometric Progression

A geometric progression is a specific type of sequence of numbers where each term is derived from the previous one by multiplying by a fixed, non-zero number called the common ratio.
This type of sequence represents exponential growth or decay due to the multiplicative nature of its development.

• The sequence: 144, -12, 1, -\( \frac{1}{12} \), demonstrates geometric progression through alternate signs and decreasing values.
• In this sequence, the common ratio \(-\frac{1}{12}\) leads to successive division and alternation of signs, representing decay and flips in positivity.

Geometric progressions are found across diverse fields like finance, science, and computer science, showcasing their wide applicability. Understanding these sequences is essential to grasping patterns that grow or decay exponentially.