Problem 17 - College Algebra

Question

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 3,12,48,192, \dots $$

Step-by-Step Solution

Answer

The seventh term of the given geometric sequence is 12288.

Step 1: Determine the First Term and the Common Ratio

The first term \(a_{1}\) is the first number in the sequence, which is 3. The common ratio \(r\) can be determined by dividing any term by its preceding term. For example, \(r = 12/3 = 4\). So, \(a_{1} = 3\) and \(r = 4\).

Step 2: Apply the formula for the nth term of a geometric sequence

Substitute the first term \(a_{1}\), the common ratio \(r\), and the term number \(n\) into the formula \(a_{n} = a_{1} \times r^{(n-1)}\). For the 7th term \(n = 7\), the formula becomes: \(a_{7} = 3 \times 4^{(7-1)}\).

Step 3: Calculate the Value

Compute the value of the expression to find the seventh term, \(a_{7} = 3 \times 4^{6}\) which equals 12288.

Key Concepts

General Term Formula

In a geometric sequence, each term is derived from the previous one by multiplying it by a constant number called the "common ratio." To find any term in such a sequence, we use the general term formula. The formula for the nth term, denoted as \(a_n\), of a geometric sequence is given by:
\[a_n = a_1 \times r^{(n-1)}\]This formula allows us to determine any term in the sequence without writing out all the preceding terms. The variables in the formula are:

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

Using this formula can save you a lot of time, particularly for finding terms further down the sequence, like the 50th or 100th term.
Understanding this formula is critical for efficiently handling geometric sequences.

Common Ratio

The common ratio \(r\) is a fundamental part of any geometric sequence. It is the factor by which we multiply each term to get to the next one.
To find the common ratio, you divide any term of the sequence by its previous term. For instance, from the sequence
\[3, 12, 48, 192, \ldots\]we calculate the common ratio as follows:
\[r = \frac{12}{3} = 4\]This calculation tells us that each term is 4 times the previous term. The common ratio can be any non-zero number, including fractions and negative numbers.
Recognizing this ratio is key to understanding how the sequence progresses.

Nth Term

The "nth term" refers to any term within a sequence, located at the nth position. In our context, it's the position number for which we aim to calculate the specific term's value. The formula to find this is

• \(a_n = a_1 \times r^{(n-1)}\)

where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number we want to determine.
For example, if you want the 7th term in a geometric sequence where the first term is 3 and the common ratio is 4, you would substitute these values and \(n = 7\) into the nth term formula:
\[a_7 = 3 \times 4^{(7-1)}\]This approach allows seamless computation of any term's value without calculating the intermediate terms.

Sequence Calculation

Calculating terms within a geometric sequence relies on a structured approach using the general formula and understanding of its components.
For our specific sequence \[3, 12, 48, 192, \dots\]To find the 7th term, \(a_7\), follow this process:

• Identify where \(a_1 = 3\), the sequence's first term.
• Determine the common ratio \(r = 4\).
• Plug these and \(n = 7\) into the formula \(a_n = a_1 \times r^{(n-1)}\).
• Calculate \(a_7 = 3 \times 4^6 = 12288\).

By following these steps, you ensure accurate results. Calculations for any other nth term in the sequence would follow a similar pattern, adapting the term number and repeating the substitution and exponential calculation.