Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 90

9장, 문제 90

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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,...