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Chapter 11: Problem 74

Explain how to find the general term of an arithmetic sequence.

Short Answer

Expert verified
The general term of an arithmetic sequence is determined with the formula \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \( d \) is the common difference and \( n \) is the term number. Calculate the first term and the common difference and replace in the formula along with \( n \).

Step by step solution

01

Understand the Formula

Begin with the formula; \( a_n = a_1 + (n - 1)d \). Here, \( a_n \) represents the \( n^{th} \) term in the sequence, \( a_1 \) is the first term of the sequence, \( n \) is the term number and \( d \) is the common difference between each term.
02

Identify the First Term

The first term (\( a_1 \)) of the sequence is usually directly given in the problem or sometimes can be determined by examining the pattern.
03

Identify the Common Difference

The common difference (\( d \)) is the constant difference between successive terms of the sequence. Calculate this value by subtracting an earlier term from a later term.
04

Define the Position of the Term

Recognize \( n \) as the position or term number of the sequence. For example, for finding the 5th term, \( n = 5 \). For the general term, leave \( n \) as is.
05

Apply the Formula

Input identified \( a_1 \) value and \( d \) value into the formula. Keep \( n \) as is, this will give us the formula for the general term of the given sequence.

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Most popular questions from this chapter

In Exercises \(105-108\), 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 \(\frac{1}{2}\)

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. You are offered a job that pays \(\$ 30,000\) for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year \(2,\) your salary will be 1.05 times what it was in the previous year. What can you expect to earn in your sixth year on the job?

You are dealt one card from a 52-card deck. Find the probability that you are dealt a red 7 or a black \(8 .\)

The probability that South Florida will be hit by a major hurricane (category 4 or 5 ) in any single year is \(\frac{1}{16}\) (Source: National Hurricane Center) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years? c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

Use the formula for the value of an annuity to solve Exercises 77鈥�84. Round answers to the nearest dollar. To save money for a sabbatical to earn a master's degree, you deposit \(\$ 2000\) at the end of each year in an annuity that pays \(7.5 \%\) compounded annually. a. How much will you have saved at the end of five years? b. Find the interest.
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