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 of the sequence, which is given as $a{1}_{}=11$.
2. Recognize that the formula for the nth term of the sequence is $a{n}_{}=a{1}_{}{3}^{n-1}$, where 3 is the common ratio.
3. Substitute $7$ for $n$ in the formula to find the seventh term: $a{7}_{}=113^{7-1}$.
4. Simplify the exponent by calculating $7\; -\; 1\; =\; 6$, so the expression becomes $a{7}_{}=\; 11\; imes\; 3^\{6\}$.
5. To find the seventh term, multiply 11 by $3^\{6\}$. (You can calculate $3^\{6\}$ first, then multiply by 11.)

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. In this problem, the sequence 11, 33, 99, 297,... is geometric with a common ratio of 3.

General Term Formula for Geometric Sequences

The nth term of a geometric sequence can be found using the formula 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 direct calculation of any term without listing all previous terms.

Exponentiation in Sequences

Exponentiation involves raising a number to a power, which in sequences represents repeated multiplication. In the formula an = a₁ * r^(n-1), the exponent (n-1) indicates how many times the common ratio is multiplied, crucial for finding terms like the seventh term.