Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1) to find the seventh term of the sequence 11, 33, 99, 297,...

Verified step by step guidance

1. Identify the first term \( a_1 \) of the sequence, which is given as 11.
2. Recognize that the sequence follows a geometric pattern with a common ratio. Calculate the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{33}{11} = 3 \).
3. Use the formula for the nth term of a geometric sequence: \( a_n = a_1 \cdot r^{(n-1)} \).
4. Substitute the known values into the formula to find the seventh term: \( a_7 = 11 \cdot 3^{(7-1)} \).
5. Simplify the expression by calculating \( 3^6 \) and then multiplying the result by 11 to find \( a_7 \).

Key Concepts

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the sequence 11, 33, 99, 297 has a common ratio of 3, as each term is three times the previous one.

Explicit Formula for Geometric Sequences

The explicit formula for a geometric sequence is given by an = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. This formula allows you to calculate any term in the sequence directly without needing to find all preceding terms.

Finding Specific Terms

To find a specific term in a sequence using the explicit formula, substitute the values of a₁, r, and n into the formula. For example, to find the seventh term of the sequence, you would set a₁ = 11, r = 3, and n = 7, and then calculate a₇ = 11 * 3^(7-1).