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Chapter 14: Problem 2

Find the common difference for each arithmetic sequence. $$3,8,13,18, \dots$$

Short Answer

Expert verified
The common difference of the arithmetic sequence is 5.

Step by step solution

01

Identify the first two terms

The first two terms in the sequence are 3 and 8.
02

Subtract the first term from the second

Subtract the first term from the second term to find the common difference. In other words, to find the common difference, subtract 3 (the first term) from 8 (the second term). This gives \(8 - 3 = 5\).
03

Verify the common difference

To verify that this is the common difference, subtract the second term (8) from the third term (13). This also results in 5 (i.e., \(13 - 8 = 5\)), confirming that the common difference for the given arithmetic sequence is 5.

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence, or arithmetic progression, is a series of numbers in which each term increases or decreases by a constant amount called the common difference. Imagine placing beads at equal distances on a string; each bead represents a term, and the equal gap between them represents the common difference.

Mathematically, an arithmetic sequence can be represented by the formula for the nth term: \( a_n = a_1 + (n-1)d \)
where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference.
Common Difference
The common difference is what makes an arithmetic sequence tick. It's the consistent interval that separates each term in the sequence. To identify it, you simply subtract one term from its successor. Think of it as the heartbeat of the sequence鈥攔egular and predictable.

For the sequence \( 3, 8, 13, 18, ... \), the common difference is obtained by computing \( 8 - 3 = 5 \). This value is not just a one-time calculation; every consecutive term difference confirms the sequence's arithmetic nature.
Sequence Patterns
Unraveling sequence patterns is like decoding a secret message. In arithmetic sequences, the pattern is linear and straightforward, marked by the additive rhythm of the common difference. It's crucial to spot this pattern early, as it streamlines the process of predicting future terms and identifying the sequence's nature.

Understanding the pattern allows for efficient problem-solving. For instance, if a sequence begins with \( 3, 8, 13... \), you can predict the next terms (18, 23, etc.) by continuing to add the common difference.
Algebraic Expressions
Algebraic expressions are the symbols and letters that stand in for numbers and operations in the algebraic world. Think of algebraic expressions as a sort of mathematical shorthand. For sequences, the expressions often contain variables that represent the position of a term in the sequence (n) and the pattern of the sequence itself, like the common difference (d).

The expression for the nth term of our arithmetic sequence, \( a_n = a_1 + (n-1)d \), paints with broad strokes how each term is linked to its position (n) and the common difference (d). It's a tool for predicting any term in the sequence without having to list them all out.

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Most popular questions from this chapter

Subtract: \(\frac{x}{x+3}-\frac{x+1}{2 x^{2}-2 x-24}\). (Section 7.4, Example 7)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}$$

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(10-5+\frac{5}{2}-\frac{5}{4}+\cdots=\frac{10}{1-\frac{1}{2}}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=0}^{6}(-1)^{i}(i+1)^{2}=\sum_{i=1}^{7}(-1) j^{2}$$
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