Problem 23 - Precalculus: Mathematical for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$2,6,18,54, \dots$$

Step-by-Step Solution

Answer

Common ratio is 3; fifth term is 162; nth term is \(a_n = 2 \times 3^{n-1}\).

Step 1: Identify the first term

The given sequence is a geometric sequence, and the first term, denoted by \(a_1\), is the first number in the sequence. In this case, \(a_1 = 2\).

Step 2: Determine the common ratio

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio \(r\). To find \(r\), divide the second term by the first term: \(r = \frac{6}{2} = 3\).

Step 3: Calculate the fifth term

The fifth term \(a_5\) of a geometric sequence can be found using the formula \(a_n = a_1 \times r^{n-1}\). Plug in the values: \(a_5 = 2 \times 3^{5-1} = 2 \times 81 = 162\).

Step 4: General expression for the nth term

The general formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \times r^{n-1}\). For this sequence, the expression is \(a_n = 2 \times 3^{n-1}\).

Key Concepts

Common Ratio

In the realm of geometric sequences, the common ratio is a pivotal concept. It describes the particular constant by which you multiply each term to get to the next term in the sequence. This number helps define the sequence not just in the short term, but throughout all of its terms.
For example, consider the sequence given: 2, 6, 18, 54, ... To unearth the common ratio, take any term (after the first) and divide it by its preceding term. In this sequence, dividing the second term (6) by the first term (2) gives us the common ratio: \[ r = \frac{6}{2} = 3 \]

• Every term after the first is thrice the previous one.
• This constancy is what links all terms to form a predictable pattern.

Recognizing this ratio is crucial because it not only characterizes the sequence but also aids in anticipating any desired term using the right formulas.

Nth Term

The nth term expression in a geometric sequence grants us a powerful tool, enabling us to find any term in the sequence swiftly. This expression encapsulates the rule that governs the entire sequence.
The general formula for finding the nth term in a geometric sequence is:\[ a_n = a_1 \times r^{n-1} \]

• Here, \(a_1\) represents the first term (in our case, 2), \(r\) is the common ratio (3), and \(n\) symbolizes the position of the term you seek.
• By plugging values into this formula, you can calculate any term of the sequence without enumerating all previous terms.

For our sequence:\[ a_n = 2 \times 3^{n-1} \]This flexibility is incredibly advantageous for managing sequences with many terms, allowing one to rapidly leap to any position in the sequence.

Fifth Term

Understanding how to find specific terms in a sequence, like the fifth term, hones one's skills in utilizing the general formula of sequences. To pinpoint the fifth term, you apply the nth term formula we previously discussed.
Let's see how this works with our sequence:

• Utilize the formula: \[ a_n = a_1 \times r^{n-1} \]
• For the fifth term, set \(n = 5\):\[ a_5 = 2 \times 3^{5-1} \]
• Calculating further:\[ = 2 \times 3^4 = 2 \times 81 = 162 \]

This results show that the fifth term in this specific sequence is 162. By mastering this technique, finding any specific term in the sequence becomes straightforward and efficient.