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Chapter 14: Problem 62

Explain how to write terms of a sequence if the formula for the general term is given.

Short Answer

Expert verified
To generate sequence terms from a given general term formula, substitute the term number for \(n\) in the formula and calculate the result. Continue this for as many terms as required.

Step by step solution

01

Understanding the Concept

The first step is to understand what a sequence is and what the general term represents. In mathematics, a sequence is a list of numbers in a specific order. This order is given by a rule that is defined by the general term. The general term formula serves as a blueprint to generate all possible terms in the sequence.
02

Identify the General Term

Suppose the general term of a sequence is given as \(a_n = 2n + 3\). The \(n\) represents the term number, and \(a_n\) represents the nth term of the sequence.
03

Generating Terms Using the General Term

To generate terms of the sequence, simply substitute the term number for \(n\). For example, to find the first term of the sequence, substitute 1 in place of \(n\) in the formula: \(a_1 = 2(1) + 3 = 5\). Similarly, to generate the second term, substitute 2 for \(n\): \(a_2 = 2(2) + 3 = 7\). The process is repeated to generate as many terms in the sequence as required.
04

Noting the Pattern

As more and more terms are generated, a pattern may emerge. Identifying and understanding the underlying pattern is also a valuable skill in working with sequences.

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Most popular questions from this chapter

Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely what terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}\).

Explain how to find \(n !\) if \(n\) is a positive integer.
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