Problem 33 - Precalculus Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$3,3^{5 / 3}, 3^{7 / 3}, 27, \dots$$

Step-by-Step Solution

Answer

The common ratio is \(3^{2/3}\); the fifth term is \(3^{11/3}\); the nth term is \(3^{(2n+1)/3}\).

Step 1: Identify the First Term

The first term of the sequence is given as 3.

Step 2: Calculate the Common Ratio

To find the common ratio \(r\), divide the second term by the first term. The second term is \(3^{5/3}\) and the first term is 3. Thus, \(r = \frac{3^{5/3}}{3} = 3^{2/3}\).

Step 3: Verify the Ratio with the Third Term

Verify the common ratio by checking it against the third term. The third term is \(3^{7/3}\) and should be equal to the second term \(3^{5/3}\) times \(r\), \(3^{7/3} = 3^{5/3} \times 3^{2/3}\), which verifies the common ratio \(r = 3^{2/3}\).

Step 4: Calculate the Fifth Term

To calculate the fifth term, use the formula for the general term of a geometric sequence \(a_n = a \cdot r^{n-1}\). For the fifth term, \(n=5\), where \(a=3\) and \(r=3^{2/3}\), so \(a_5 = 3 \cdot (3^{2/3})^{4} = 3 \cdot 3^{8/3} = 3^{11/3}\).

Step 5: Find the General Formula for the nth Term

The general formula for a geometric sequence is \(a_n = a \cdot r^{n-1}\). Substituting the values of \(a = 3\) and \(r = 3^{2/3}\), we get \(a_n = 3 \cdot (3^{2/3})^{n-1} = 3^{1+ 2(n-1)/3} = 3^{\frac{3+2n-2}{3}} = 3^{\frac{2n+1}{3}}\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the factor that you multiply a term by to get the next term in the sequence. It's a vital component that dictates how the sequence progresses. To find the common ratio, denoted as \(r\), you look at consecutive terms in the sequence.
If you take the sequence from the exercise, \(3, 3^{5/3}, 3^{7/3}, 27, \ldots\), you can find \(r\) by dividing the second term by the first term. Specifically:

• First term: \(3\)
• Second term: \(3^{5/3}\)
• Common Ratio: \(r = \frac{3^{5/3}}{3} = 3^{2/3}\)

Understanding the common ratio helps you grasp how each term relates to its predecessor and successor in a geometric sequence.

nth Term Formula

The nth term formula is a powerful tool that allows you to find any term in a geometric sequence without having to write out all the previous terms. It is given by the expression \(a_n = a \cdot r^{n-1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number you want to find.
In our sequence, the first term \(a = 3\) and the common ratio \(r = 3^{2/3}\). Plug these into the formula to get:

• \(a_n = 3 \cdot (3^{2/3})^{n-1} = 3^{\frac{2n+1}{3}}\)

This formula allows you to compute any term in the sequence efficiently, making it an essential part of understanding and working with geometric sequences.

Sequence Verification

Verifying a sequence ensures that your calculated common ratio is indeed correct and that the progression of terms matches expectations. This involves checking if the subsequent terms can be generated consistently using the identified common ratio.
In the exercise:

• Second term: \(3^{5/3}\)
• Third term: \(3^{7/3}\) (calculated as \(3^{5/3} \times 3^{2/3}\))

By performing these multiplications, if the results match the given sequence, the ratio \(r = 3^{2/3}\) is verified as correct. It's always a good idea to check a couple of terms to ensure your sequence is valid as overlooking this might lead to mistakes.

Term Calculation

Term calculation in a geometric sequence involves using the nth term formula to determine specific terms you're interested in. For our specific task:

• We calculated the fifth term, \(a_5\), in the sequence using the formula:
• \(a_5 = 3\cdot (3^{2/3})^{4} = 3 \cdot 3^{8/3} = 3^{11/3}\)

With this process, you derive any subsequent term in the sequence without manually multiplying the common ratio repeatedly.
This method simplifies the calculation significantly, especially when working with larger indices or more complex sequences, while reinforcing the underlying geometric progression principle.