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Chapter 11: Problem 26

Find the missing term of each arithmetic sequence. \(203, \square, 1117, \ldots\)

Short Answer

Expert verified
The missing term of the arithmetic sequence is 660.

Step by step solution

01

Identify the Known Terms

First, identify the known terms of the sequence. In this case, the first term \(a_1\) is 203 and the third term \(a_3\) is 1117.
02

Find the Common Difference

Since the difference between consecutive terms in an arithmetic sequence is constant, we can express the third term as follows: \(a_3 = a_1 + 2d\), where \(d\) is the common difference. Substituting the known values, we get: \(1117 = 203 + 2d\). Solving for \(d\), we find that \(d = 457\).
03

Find the Missing Term

Now, to find the second term or the missing term \(a_2\), we can use the formula of an arithmetic sequence: \(a_2 = a_1 + d\). Substituting the known values, we get: \(a_2 = 203 + 457\). Therefore, \(a_2 = 660\).

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Key Concepts

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

Common Difference
In an arithmetic sequence, the common difference is a key element. It refers to the consistent difference between consecutive terms in the sequence. This characteristic is what makes arithmetic sequences distinct. For example, if you observe the numbers in the sequence closely, you'll see the same number being added or subtracted to get from one term to the next. In the exercise above, we're given the first term as 203 and the third term as 1117. By using these, we can determine the common difference.To find the common difference \(d\), use the formula:\[ a_3 = a_1 + 2d \]Substitute the known values:\[ 1117 = 203 + 2d \]This equation helps us solve for \(d\). Simplify and rearrange, and you end up with:\[ d = 457 \]Thus, the common difference \(d\) is 457 in this sequence.
Arithmetic Sequence Formula
The arithmetic sequence formula is fundamental when working with sequences. It gives us the ability to find any term in the sequence when we know certain details. The basic formula is:\[ a_n = a_1 + (n-1)d \]Where:
  • \(a_n\) represents the \(n\)-th term you want to find.
  • \(a_1\) is the first term in the sequence.
  • \(d\) stands for the common difference.
  • \(n\) is the term number.
Knowing this formula allows you to substitute given values and solve for any term. For the sequence we are examining, once the common difference \(d = 457\) is established, this formula can be utilized to pinpoint other terms in the sequence.
Finding Missing Terms in Sequences
Finding missing terms in an arithmetic sequence is like solving a puzzle. Once you have the steps down, it's all about applying them.To find a missing term, first gather the known terms and the common difference, \(d\). In our exercise, the first term is 203, the third is 1117, and we calculated that the common difference \(d\) is 457.The missing term is the second one, \(a_2\). To find \(a_2\) using the arithmetic sequence formula, add the common difference to the first term:\[ a_2 = a_1 + d \]Substitute the known values:\[ a_2 = 203 + 457 \]Therefore, the missing term \(a_2\) is 660.By following these steps and formulas, any missing term in a sequence becomes easy to calculate.

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Most popular questions from this chapter

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