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,...

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Identify the first term of the sequence, which is given as

a 1 = 11

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.

Substitute

7

for

n

in the formula to find the seventh term:

a 7 = 11 3^{7 - 1}

Simplify the exponent by calculating

7 - 1 = 6

, so the expression becomes

= 11 imes 3^{6}

To find the seventh term, multiply 11 by

3^{6}

. (You can calculate

3^{6}

first, then multiply by 11.)

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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