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Chapter 9: Problem 22

Find a formula for the \(n\)th term of the sequence. The sequence \(2,6,10,14,18, \ldots\)

Short Answer

Expert verified
The formula is \( a_n = 4n - 2 \).

Step by step solution

01

Identify the Pattern

First, let's observe the sequence: 2, 6, 10, 14, 18, .... Notice that each term increases by 4 from the previous term (e.g., 6 - 2 = 4, 10 - 6 = 4). This is an arithmetic sequence.
02

Determine General Form

In an arithmetic sequence, each term can be represented as: \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference. Here, \( a_1 = 2 \) and \( d = 4 \).
03

Plug into the Formula

Substituting the values into the general form: \( a_n = 2 + (n-1) imes 4 \). Simplify it to find the formula in terms of \( n \).
04

Simplify the Formula

Simplify the equation: \( a_n = 2 + 4(n-1) \) to \( a_n = 2 + 4n - 4 \). Further simplification gives \( a_n = 4n - 2 \). This is the formula for the \(n\)th term.

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Key Concepts

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

Understanding the Sequence Formula
An arithmetic sequence is a series of numbers where each term is generated by adding a fixed number to the previous term.
This fixed number is known as the 'common difference'. In the given sequence: 2, 6, 10, 14, 18,..., we identify this as an arithmetic sequence because the difference between consecutive terms remains constant.

The formula for finding any term in this sequence, denoted as the -th term or \(a_n\), is:
  • \( a_n = a_1 + (n-1) \times d \)
Where:
  • \( a_1 \) is the first term of the sequence.
  • \( d \) is the common difference between the terms.
  • \( n \) is the term number you want to find.
Understanding this formula is key to identifying any term of the sequence with ease. It gives you a systematic method to calculate what you need without listing all the terms.
Exploring the Common Difference
The common difference is crucial in arithmetic sequences. It's the consistent amount added to each term to get to the next one.
For the sequence 2, 6, 10, 14, 18,..., the common difference \( d \) can be calculated by subtracting a term from the previous term.

Let's break this down:
  • From 2 to 6: \( 6 - 2 = 4 \)
  • From 6 to 10: \( 10 - 6 = 4 \)
  • And so on...
Hence, the common difference \( d \) is consistently \( 4 \).

This stable increase by the common difference ensures the arithmetic nature of the sequence. It acts as the foundational step in identifying and using the sequence formula. Understanding the common difference helps you immediately spot the pattern and predict future terms.
Term Identification Techniques
In arithmetic sequences, identifying a specific term involves applying the sequence formula directly. If you need, say, the 5th term, you can use the formula.
With the sequence formula \( a_n = a_1 + (n-1) \times d \), determine the needed term by plugging in the values of \( a_1 \), \( n \), and \( d \).

Let's illustrate:
  • For \( a_1 = 2 \), \( n = 5 \), and \( d = 4 \), substitute these into the formula:
  • \( a_5 = 2 + (5-1) \times 4 \)
  • Simplify to get: \( a_5 = 2 + 16 = 18 \)
Thus, the 5th term is 18.

This practical application shows how easily you can pinpoint any term in the sequence without listing all terms sequentially. It's a powerful method that saves time and effort, especially for larger sequences.

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Most popular questions from this chapter

Make up a geometric series \(\sum a r^{n-1}\) that converges to the number 5 if a. \(a=2\) b. \(a=13 / 2\)

Prove that a sequence \(\left\\{a_{n}\right\\}\) converges to 0 if and only if the sequence of absolute values \(\left\\{\left|a_{n}\right|\right\\}\) converges to 0.

In each of the geometric series, write out the first few terms of the series to find \(a\) and \(r\), and find the sum of the series. Then express the inequality \(|r|<1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$

Make up an infinite series of nonzero terms whose sum is a. 1 b. \(-3 \quad\) c. 0

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L ?\) b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$a_{n}=\frac{\ln n}{n}$$
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