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Chapter 17: Problem 362

Find the twelfth term of the arithmetic sequence \(2,5,8, \ldots\)

Short Answer

Expert verified
The twelfth term of the arithmetic sequence is \(a_{12} = 35\).

Step by step solution

01

Identify the known values

We know that the first term \(a_1 = 2\), the common difference \(d = 3\), and the term number we want to find is \(n = 12\).
02

Apply the arithmetic sequence formula

We will use the arithmetic sequence formula to find the twelfth term: \(a_n = a_1 + (n-1)d \)
03

Plug in the known values

Now, we will plug in the values of \(a_1\), \(d\), and \(n\) into the formula: \(a_{12} = 2 + (12 - 1) \cdot 3\)
04

Simplify the expression

Next, we will simplify the expression: \(a_{12} = 2 + (11) \cdot 3\)
05

Simplify further and find the twelfth term

Finally, we will multiply and add the values to get our answer: \(a_{12} = 2 + 33\) \(a_{12} = 35\) The twelfth term of the arithmetic sequence is 35.

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Key Concepts

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

Arithmetic Sequence Formula
An arithmetic sequence is a series of numbers in which the difference between any two consecutive terms is always the same and is known as the common difference. To find any term in an arithmetic sequence, we can use a specific formula:
\the arithmetic sequence formula is expressed as
\the arithmetic sequence formula is expressed as \[ a_n = a_1 + (n - 1)d \].
Here, \( a_n \) represents the \( n^{th} \) term of the sequence, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. This formula provides a straightforward method to calculate the value of any term within the sequence without having to list all the preceding terms.
Common Difference
The common difference is a key feature of an arithmetic sequence. It is the constant amount that each term increases or decreases from the previous term. To identify the common difference, we can subtract any term from the following term in the sequence. In the sequence \(2, 5, 8, \ldots\), for instance, the common difference is found by subtracting the first term from the second, resulting in \(5 - 2 = 3\). This common difference is essential for using the arithmetic sequence formula effectively. Understanding the role of the common difference helps students grasp how arithmetic sequences are built and how they progress.
Sequence Term Calculation
When it comes to calculating a specific term in an arithmetic sequence, the process involves plugging known values into the arithmetic sequence formula. Following a systematic approach ensures accuracy and understanding:
    \t
  • Identify the first term \( a_1 \), the common difference \( d \), and the term number \( n \) you wish to find.
    • \t
  • Apply the arithmetic sequence formula.
    • \t
  • Insert the known values into the formula.
    • \t
  • Simplify the expression by performing the multiplication and addition.
    • \t
  • The result is the value of the term you were seeking.
Using this method, you can find any term in an arithmetic sequence without having to write out the entire sequence. This is a powerful tool for solving problems quickly, especially when dealing with sequences that have a large number of terms.

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Most popular questions from this chapter

The fourth term of a geometric progression is \(1 / 2\) and the sixth term is \(1 / 8\). Find the first term and the common ratio.

1f \(a^{2}, b^{2}, c^{2}\) are in arithmetic progression, show that \(b+c\), \(c+a, a+b\) are in harmonic progression.

Insert 20 arithmetic means between 4 and 67 .

If the first term of a geometric progression is 9 and the common ratio is \(-2 / 3\) find the first five terms.

If the 6 th term of an arithmetic progression is 8 and the 11 th term is \(-2\), what is the 1 st term? What is the common difference?
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