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Chapter 11: Problem 442

How many terms are in the finite arithmetic sequence \(12,20,28, \ldots, 172 ?\)

Short Answer

Expert verified
The sequence has 21 terms.

Step by step solution

01

Identify Sequence Components

The given arithmetic sequence is: 12, 20, 28, ..., 172. From this sequence, we can identify the first term \(a_1 = 12\) and the common difference \(d = 20 - 12 = 8\). The last term in the sequence is \(172\).
02

Use the Arithmetic Sequence Formula

The \(n\)-th term of an arithmetic sequence can be found by the formula: \(a_n = a_1 + (n-1) \cdot d\). We know \(a_n = 172\), \(a_1 = 12\), and \(d = 8\). Substitute these into the arithmetic sequence formula: \[172 = 12 + (n-1) \cdot 8.\]
03

Solve for \(n\)

Rearrange and solve the equation from Step 2 for \(n\):\[172 = 12 + (n-1) \cdot 8.\] Simplify: \[172 - 12 = (n-1) \cdot 8\] \[160 = (n-1) \cdot 8.\] Divide both sides by 8: \[\frac{160}{8} = n-1\] \[20 = n-1.\] Add 1 to both sides: \[n = 21.\]
04

Conclusion

There are 21 terms in the sequence. The last term \(172\) is achieved when \(n = 21\), which confirms that there are 21 terms in the finite arithmetic sequence.

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Key Concepts

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

Common Difference
A key feature of an arithmetic sequence is the common difference. This is the difference between consecutive terms in the sequence, and it remains constant throughout the sequence. Think of it as the "step" you take to move from one term to the next.
In the given sequence, the first few terms are 12, 20, and 28. We calculate the common difference, denoted as \(d\), by subtracting the first term from the second term:
  • \(d = 20 - 12 = 8\)
This tells us that each term is 8 more than the term before it. Understanding the common difference helps us to quickly identify how the sequence grows and aids in writing the sequence formula.
Sequence Formula
The sequence formula allows us to find any term in the arithmetic sequence without having to list out all the terms. The formula for the \(n\)-th term \(a_n\) in an arithmetic sequence is given as:
  • \(a_n = a_1 + (n-1) \cdot d\)
Here, \(a_1\) is the first term and \(d\) is the common difference.
For our sequence, you can find the 21st term by plugging into the formula:
  • \(a_n = 12 + (21-1) \cdot 8 = 12 + 160 = 172\)
Using the formula is like having a shortcut to find any term's value without calculating all previous terms. Knowing this formula is essential for tackling any arithmetic sequence problem efficiently.
Finite Sequence
An arithmetic sequence can be either finite or infinite. In a finite sequence, there is a definite number of terms, as opposed to an infinite sequence that goes on forever. Recognizing whether a sequence is finite is crucial when solving problems regarding its properties.
In our exercise, the sequence starts at 12 and ends at 172, making it a finite sequence. We used the sequence formula to solve for the number of terms \(n\) by working from the provided final term \(a_n = 172\).
This means that:
  • There are 21 total terms in this sequence.
Understanding that the sequence is finite helps in determining how to use techniques and formulas appropriately to reach an answer.

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