Problem 16 - Algebra and Trigonometry

Question

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{f_{n}\right\\}=\left\\{3^{2 n}\right\\} $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 9. The first four terms are 9, 81, 729, and 6561.

Step 1 Title - Identify the General Term

The general term for the sequence is given by \( f_n = 3^{2n} \). This represents the nth term of the sequence.

Step 2 Title - Find the First Four Terms

Substitute \( n = 1, 2, 3, 4 \) into the general term to find the first four terms: For \( n = 1 \), \( f_1 = 3^{2 \times 1} = 9 \)For \( n = 2 \), \( f_2 = 3^{2 \times 2} = 81 \)For \( n = 3 \), \( f_3 = 3^{2 \times 3} = 729 \)For \( n = 4 \), \( f_4 = 3^{2 \times 4} = 6561 \).

Step 3 Title - Determine the Common Ratio

To determine the common ratio, divide each term by the previous term: \( f_2/f_1 = 81/9 = 9 \)\( f_3/f_2 = 729/81 = 9 \)\( f_4/f_3 = 6561/729 = 9 \)The ratio is constant and equal to 9, confirming the sequence is geometric.

Step 4 Title - Conclusion

Since the ratio between consecutive terms is constant, the given sequence is geometric with a common ratio of 9.

Key Concepts

common ratio

In a geometric sequence, the common ratio is the factor that you multiply each term by to get the next term. You can find it by dividing any term in the sequence by the previous term. In this exercise, we found it by calculating the ratio between each consecutive pair of terms:

• First, we identified the terms: 9, 81, 729, and 6561.
• Then, we divided each term by the one before it. For example, 81 divided by 9 gives us 9.

Thus, the common ratio in this sequence is 9. It's helpful to remember that if the common ratio is constant for all pairs of terms, the sequence is geometric.

sequence terms

A sequence is just a list of numbers in a specific order. In a geometric sequence, each term is generated by multiplying the previous term by a fixed number called the common ratio.

• For example, if we start with the first term of our sequence, which is 9, we can find the next term by multiplying 9 by the common ratio, which we found to be 9.
• So, the next term is 81, followed by 729 and then 6561.

These specific numbers form the sequence given by the expression \( 3^{2n} \) when substituting \( n = 1, 2, 3, \) and \( 4 \). It's important to remember that each term depends on its position in the sequence, denoted by 'n'.

general term

The general term of a sequence gives us a formula to find any term in the sequence without listing all the previous ones. For a geometric sequence, the general term can be written as:

• \( f_n = ar^{(n-1)} \), where 'a' is the first term and 'r' is the common ratio.

For the sequence in the exercise, the general term is given by \( f_n = 3^{2n} \). This tells us how to find any term based on its position 'n'.

• To find the first term, substitute \( n = 1 \): \( 3^{2 \times 1} = 9 \).
• To find the second term, substitute \( n = 2 \): \( 3^{2 \times 2} = 81 \).

By using the general term, you can easily find any term without calculating all previous ones, making it a powerful tool for understanding sequences.