Problem 42 - Algebra 2

Question

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 5th term

Step-by-Step Solution

Answer

The 5th term of the given geometric sequence is 768.

Step 1: Identify Common Ratio

Analyze the sequence and determine the common ratio r. The common ratio is obtained by dividing any term by its preceding term, which in this case is 12/3 = 4.

Step 2: Apply Formula to Find 5th Term

Apply the formula for the nth term of a geometric sequence: \(a_n = a_1 * r^{(n-1)}\). The first term \(a_1\) is 3, the common ratio r is 4, and the term number n is 5. Substituting these values into the formula gives \(a_5 = 3 * 4^{(5-1)}\).

Step 3: Calculate 5th Term

Do the calculation: \(a_5 = 3 * 4^4 = 3 * 256 = 768\). The fifth term of the sequence is 768.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, one of the key concepts to grasp is the common ratio. This is a critical component that distinguishes geometric sequences from other types of sequences. Simply put, the common ratio is the factor by which each term in the sequence is multiplied to get the next term. To find the common ratio, you can take any term in the sequence and divide it by its preceding term.

For example, in the sequence given: 3, 12, 48, 192, we calculate the common ratio as follows:

• Divide 12 (the second term) by 3 (the first term) to get 4.
• This indicates that each term is multiplied by 4 to get the next term in this sequence.

With the ratio, we can easily predict future terms in the sequence.

What is the nth Term Formula?

The nth term formula is a powerful tool that helps in finding any term in a geometric sequence without listing out all the terms one by one. The formula is: \[ a_n = a_1 \times r^{(n-1)} \]where:

• \( a_n \) is the nth term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) represents the common ratio.
• \( n \) is the position of the term in the sequence.

This formula saves time and effort, especially with sequences where calculating each term manually is cumbersome. By plugging in the known values, you can find the specific term you’re interested in. For example, using 3 as the first term and 4 as the common ratio, you can accurately determine the 5th term in the sequence without needing to compute each term leading up to it.

Grasping Geometric Progression

A geometric progression, sometimes referred to as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the preceding one by a fixed, non-zero number called the common ratio. In simpler terms, it’s a series of numbers that multiply by the same factor each time.

Features of a geometric progression include:

• The sequence has a constant ratio among consecutive terms.
• It can either increase or decrease depending on whether the common ratio is greater than or less than one.
• Unlike arithmetic progressions, geometric progressions can grow rapidly due to exponentiation, especially when the common ratio is greater than 1.

Understanding geometric progressions is fundamental in various mathematical areas and real-life applications like calculating interest rates or population growth which involve consistent multiplying factors.