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Chapter 12: Problem 19

Find a formula for the nth term of the sequence whose first few terms are given. $$2,7,12,17,22,27, \dots$$

Short Answer

Expert verified
Answer: The formula for the nth term of the given arithmetic sequence is $$a_n = 5n - 3$$.

Step by step solution

01

Identify the pattern between the terms of the sequence

To identify the pattern, calculate the differences between consecutive terms: $$7-2=5$$ $$12-7=5$$ $$17-12=5$$ $$22-17=5$$ So, there is a constant difference of 5 between each consecutive term. This indicates that the sequence is an arithmetic sequence.
02

Find the first term and common difference

The first term in the sequence, denoted by a_1, is 2. The common difference, denoted by d, is 5 as determined in Step 1.
03

Write down the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is given by: $$a_n = a_1 + (n - 1)d$$
04

Substitute the values of a_1 and d to find the formula for the nth term

Substituting the values of a_1 = 2 and d = 5 into the formula: $$a_n = 2 + (n - 1)5$$ Simplify the expression by distributing 5 through the parentheses and combining like terms: $$a_n = 2 + 5n - 5$$ $$a_n = 5n - 3$$ The formula for the nth term of the given sequence is: $$a_n = 5n - 3$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

nth term formula
The "nth term formula" is integral in understanding arithmetic sequences. It allows us to find any term in the sequence without needing to list all the previous numbers. For an arithmetic sequence, the formula is expressed as \( a_n = a_1 + (n - 1) \, d \). Here, \( a_1 \) represents the first term, \( d \) is the common difference, and \( n \) is the term number you want to find. This formula provides a step-by-step method to trace through the pattern of the sequence and easily reach the desired term. By using this, you can save time and effort, especially with larger sequences.
common difference
The "common difference" is a crucial element of an arithmetic sequence and can be identified by subtracting one term from the consecutive term. In our sequence, \( 2, 7, 12, 17, 22, 27, \dots \), the common difference is calculated by subtracting the first term from the second: \( 7 - 2 = 5 \).
This value remains consistent throughout the sequence, which is a defining property of arithmetic sequences. Knowing the common difference helps us determine the "nth term formula" effectively. It signifies how much each successive term is increasing or decreasing as we progress through the sequence.
sequence pattern
"Sequence pattern" refers to the systematic arrangement in which the sequence is progressing. Recognizing the pattern is key to identifying the type of sequence you are dealing with. In arithmetic sequences, like the one given as \( 2, 7, 12, 17, 22, 27, \dots \), the pattern is linear, showing a consistent step from one term to the next.
By understanding the pattern, we can accurately anticipate the future terms and make calculations for the sequence effectively. The pattern provides insight and reveals the consistent rule or relationship between terms in the sequence.
consecutive terms
"Consecutive terms" are terms that appear sequentially in a sequence, each following the other without interruption. In the current sequence \( 2, 7, 12, 17, 22, 27, \dots \), each number represents a consecutive term. Identifying the relation between consecutive terms is essential to figuring out the "common difference" and developing an overall understanding of the sequence.
This concept is foundational because observing changes from one term to the next aids in determining other critical sequence properties, such as whether it's increasing, decreasing, or maintaining a specific pattern.

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Most popular questions from this chapter

Deal with prime numbers. A positive integer greater than 1 is prime if its only positive integer factors are itself and 1. For example, 7 is prime because its only factors are 7 and \(1,\) but 15 is not prime because it has factors other than 15 and 1 (namely, 3 and 5 ). Find the first five terms of the sequence. \(a_{n}\) is the number of prime integers less than \(n\)

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$-1,-\frac{1}{2}, 0, \frac{1}{2}, \dots$$

(a) Find these numbers and write them one below the next: \(11^{0}, 11^{1}, 11^{2}, 11^{3}, 11^{4}\) (b) Compare the list in part (a) with rows 0 to 4 of Pascal's triangle. What's the explanation? (c) What can be said about \(11^{5}\) and row 5 of Pascal's triangle? (d) Calculate all integer powers of 101 from \(101^{0}\) to \(101^{8}\), list the results one under the other, and compare the list with rows 0 to 8 of Pascal's triangle. What's the explanation? What happens with \(101^{9} ?\)

In Exercises \(23-30,\) show that the given sequence is geometric and find the common ratio. $$\left\\{\left(-\frac{1}{2}\right)^{n}\right\\}$$

The number of hours the average person spends listening to satellite radio in year \(n\) is approximated by the sequence \(\left\\{b_{n}\right\\},\) where \(n=1\) corresponds to 2001 and \(b_{n}=945.54+\) 73.57 ln \(n .^{*}\) Round your final answers (not your calculations) to the following questions to the nearest hour. (a) For how many hours did the average person listen to satellite radio in 2002 and \(2006 ?\) (b) How much total time did the average person spend listening to satellite radio from 2001 to 2005 (inclusive)?
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