Problem 29 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{6}=3, r=2, n=12 $$

Step-by-Step Solution

Answer

The 12th term of the sequence is 192.

Step 1: Understanding the Problem

We are given a geometric sequence where the 6th term \( a_6 = 3 \), and the common ratio \( r = 2 \). We need to find the 12th term \( a_{12} \) of this sequence.

Step 2: Recall the Formula for the nth Term

The formula for the \( n \)th term of a geometric sequence is given by:\[a_n = a_1 \, r^{n-1}\]where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 3: Find the First Term

We know \( a_6 = a_1 \, r^{5} = 3 \). Substitute \( r = 2 \) into the equation:\[a_1 \, 2^5 = 3\]Solve for \( a_1 \):\[a_1 = \frac{3}{2^5} = \frac{3}{32}\]

Step 4: Calculate the 12th Term

Now that we have \( a_1 \), use the formula for \( a_n \) to find \( a_{12} \).\[a_{12} = a_1 \, r^{11} = \frac{3}{32} \, 2^{11}\]Calculate \( 2^{11} = 2048 \).Substitute back into the equation:\[a_{12} = \frac{3}{32} \, 2048 = 3 \, \times \frac{2048}{32}\]

Step 5: Simplify the Calculation

Simplify the fraction \( \frac{2048}{32} = 64 \).So, \( a_{12} = 3 \, \times \, 64 = 192 \).

Key Concepts

Nth Term Formula in Geometric Sequences

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant, known as the common ratio. To find any term, say the nth term, of this sequence, we use the nth term formula:

• \( a_n = a_1 \times r^{n-1} \)

Here, \( a_n \) represents the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.
This formula is beneficial as it lets us find any term in the sequence without needing to list out all preceding terms.
Simply plug in the values you know, and solve for the missing term.
For example, if we know the first term is 3, the common ratio is 2, and we want the 4th term, substitute values:

• \( a_4 = 3 \times 2^{4-1} = 3 \times 8 = 24 \)

Understanding the Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply one term to get the next. Determining this ratio is crucial for working with the sequence's nth term formula.
In our exercise, the common ratio \( r = 2 \), illustrating how each term is double the previous one.
For an intuitive grasp:

• Identify any two consecutive terms in a sequence by division: \( r = \frac{a_{n}}{a_{n-1}} \).
• This works for each term, as long as the same ratio applies throughout.

The consistency of the common ratio defines the sequence, allowing us to predict future values.
Suppose in another sequence the common ratio \( r = 3 \). A second term \( a_2 = 9 \) would imply the first \( a_1 = 3 \), since \( 3 \times 3 = 9 \).

Calculating the First Term of a Sequence

The first term \( a_1 \) of a geometric sequence sets the base for all other terms.
If given an arbitrary term and the common ratio, you can retroactively find the first term.
In our original exercise, we were given the sixth term \( a_6 = 3 \) and \( r = 2 \). To find the first term, apply the nth term formula backwards:

• \( a_6 = a_1 \times 2^{5} = 3 \)
• Rearranging gives \( a_1 = \frac{3}{2^5} = \frac{3}{32} \).

This reveals that even with limited information, understanding the relationship between terms can help you uncover unknown factors.
With a first term at hand, predicting future sequence values becomes straightforward by simply applying the nth term formula.