Problem 45 - Algebra 2

Question

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 14 th term

Step-by-Step Solution

Answer

The 14th term is \(3 * 4^{13}\), or 3221225472.

Step 1: Identify the First Term and Ratio

From the provided sequence, the first term \(a_1\) is 3 and the ratio \(r\) is 4. You can find the ratio by dividing any term by its preceding term.

Step 2: Use the Geometric Sequence Formula

Next, use the formula for the nth term of a geometric sequence:\[a_n = a_1 * r^{(n-1)}\]Substitute \(a_1 = 3\), \(r = 4\), and \(n = 14\) into the formula.

Step 3: Calculate the 14th Term

Perform the calculation:\[a_{14} = 3 * 4^{(14-1)}\]This will give the 14th term of the sequence.

Key Concepts

Geometric Sequences Formula

A geometric sequence 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. One of the key aspects of understanding geometric sequences is the geometric sequences formula. This formula is used to find any term in the sequence without listing out all the terms before it.

The formula for finding the nth term of a geometric sequence is given by:

• \[ a_n = a_1 \cdot r^{(n-1)} \]

Here,:

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

Using this formula, you can efficiently calculate any term in the sequence directly, which can be much quicker than repeatedly multiplying by the common ratio.

Common Ratio

The common ratio is a vital component in the structure of a geometric sequence. It is a constant value that you multiply by in order to get from one term to the next in the sequence. Understanding the concept of the common ratio allows for a better grasp of how sequences progress.

You can find the common ratio \( r \) by taking any term in the sequence and dividing it by the previous term. For example, in the sequence: 3, 12, 48, 192, ...:

• \( r = \frac{12}{3} = 4 \)
• \( r = \frac{48}{12} = 4 \)
• \( r = \frac{192}{48} = 4 \)

As demonstrated above, the common ratio remains constant throughout the sequence, confirming that it is indeed a geometric sequence with a ratio of 4. This constant multiplication factor is what distinguishes geometric sequences from other types, such as arithmetic sequences.

Nth Term of a Sequence

The nth term of a sequence indicates a specific term's position in the sequence. For geometric sequences, determining the nth term is particularly straightforward thanks to the established formula. This allows you to find any term in the sequence without iterating through every prior term.

To find the nth term, use the formula:

• \[ a_n = a_1 \cdot r^{(n-1)} \]

In practical application, such as finding the 14th term in the sequence 3, 12, 48, 192, ..., you would substitute:

• \( a_1 = 3 \)
• \( r = 4 \)
• \( n = 14 \)

This results in:

• \[ a_{14} = 3 \cdot 4^{13} \]

This calculation provides the 14th term directly, showcasing the powerful utility of the formula for efficiently navigating through the sequence.