Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 53

9장, 문제 53

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. a_n= 2ⁿ

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

a n = 2^{n}

Recall the definitions: an arithmetic sequence has a constant difference between consecutive terms, and a geometric sequence has a constant ratio between consecutive terms.

Calculate the first few terms of the sequence by substituting values of

n

: for example,

a_1 = 2^1 = 2

a_2 = 2^2 = 4

a_3 = 2^3 = 8

Check if the difference between consecutive terms is constant: compute

a_2 - a_1

and

a_3 - a_2

. If these are equal, the sequence is arithmetic.

If the differences are not constant, check the ratio of consecutive terms: compute

\(\frac{a_2}{a_1}\)

and

\(\frac{a_3}{a_2}\)

. If these ratios are equal, the sequence is geometric and the common ratio is that value.

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주요 개념

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

Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant is called the common difference. For example, in the sequence 3, 5, 7, 9, the common difference is 2.

Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. For example, in 2, 4, 8, 16, the common ratio is 2.

General Term of a Sequence

The general term (an) of a sequence defines the nth term as a function of n. Understanding this formula helps identify the type of sequence and calculate specific terms, differences, or ratios. For example, an = 2^n defines each term as 2 raised to the power n.

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = 2n