Problem 14 - Algebra and Trigonometry

Question

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{d_{n}\right\\}=\left\\{\frac{3^{n}}{9}\right\\} $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 3. The first four terms are \( \frac{1}{3}, 1, 3, 9 \).

Step 1 - Identify the general term

The general term 'n' for the sequence is given as \(d_n = \frac{3^{n}}{9}\).

Step 2 - Find the first four terms

Substitute \(n = 1, 2, 3, 4\) into the general term to find the first four terms of the sequence. For \(n = 1\): \[d_1 = \frac{3^{1}}{9} = \frac{3}{9} = \frac{1}{3}\]For \(n = 2\): \[d_2 = \frac{3^{2}}{9} = \frac{9}{9} = 1\]For \(n = 3\): \[d_3 = \frac{3^{3}}{9} = \frac{27}{9} = 3\]For \(n = 4\): \[d_4 = \frac{3^{4}}{9} = \frac{81}{9} = 9\]

Step 3 - Verify the sequence is geometric

To check if the sequence is geometric, compute the ratio between consecutive terms and see if it remains constant. \[ \text{Common Ratio } = \frac{d_{n+1}}{d_n} First ratio: \frac{d_2}{d_1} = \frac{1}{\frac{1}{3}} = 3\] Second ratio: \[ \frac{d_3}{d_2} = \frac{3}{1} = 3\] Third ratio: \[ \frac{d_4}{d_3} = \frac{9}{3} = 3\] Since the ratio is constant (3) for all terms, the sequence is geometric.

Step 4 - State the common ratio and first four terms

The common ratio of the sequence is 3. The first four terms are: \[ \frac{1}{3}, 1, 3, 9 \]

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial concept. It is the factor by which each term in the sequence is multiplied to get the next term. For the sequence given in the exercise, the common ratio can be found by dividing any term by the preceding term. When we substitute different values for 'n', we get the terms of the sequence: \ For \( n = 1 \), \( d_1 = \frac{1}{3} \) \ For \( n = 2 \), \( d_2 = 1 \) \ For \( n = 3 \), \( d_3 = 3 \) \ For \( n = 4 \), \( d_4 = 9 \) \ To verify it is geometric, we check the ratio between consecutive terms:

• \[ \frac{d_2}{d_1} = \frac{1}{\frac{1}{3}} = 3 \]
• \[ \frac{d_3}{d_2} = \frac{3}{1} = 3 \]
• \[ \frac{d_4}{d_3} = \frac{9}{3} = 3 \]

Since the common ratio is consistent (3), the sequence is confirmed to be geometric, with a common ratio of 3.

Sequence Terms

Understanding sequence terms helps in grasping how a geometric sequence is formed. In the provided sequence defined by \( \left\{d_{n}\right\} = \left\{\frac{3^{n}}{9}\right\} \), each term is derived by substituting successive integer values for \( n \). \ The first four terms are found as follows:

• \[ d_1 = \frac{3^1}{9} = \frac{1}{3} \]
• \[ d_2 = \frac{3^2}{9} = 1 \]
• \[ d_3 = \frac{3^3}{9} = 3 \]
• \[ d_4 = \frac{3^4}{9} = 9 \]

These calculations aid in visualizing the sequence's progression. Each term is a multiple of the previous term by the common ratio. If you continue this pattern, the series will expand geometrically.

General Term

The general term of a geometric sequence is essential, as it defines the formula that generates the entire sequence. The general term for the given sequence is \( d_n = \frac{3^n}{9} \). This formula tells us how to find any term of the sequence without generating all previous terms. \ Substitute specific values for 'n' to find particular terms:

• For \( n = 1 \), \( d_1 = \frac{3^1}{9} = \frac{1}{3} \)
• For \( n = 2 \), \( d_2 = \frac{3^2}{9} = 1 \)
• For \( n = 3 \), \( d_3 = \frac{3^3}{9} = 3 \)
• For \( n = 4 \), \( d_4 = \frac{3^4}{9} = 9 \)

The general term encodes the entire sequence, making it easy to determine any term's value. This representation shows that all terms can be derived from the common ratio and initial term. Understanding the general term provides significant insight into the structure and properties of a geometric sequence.