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Chapter 13: Problem 13

Find the common difference \(d\). $$ 10,5,0,-5,-10, \ldots $$

Short Answer

Expert verified
The common difference is \( -5 \).

Step by step solution

01

Identify the First Term

Examine the sequence provided: 10, 5, 0, -5, -10, \dots The first term, denoted as \(a_1\), is 10.
02

Choose Consecutive Terms

Pick any two consecutive terms from the sequence to find the common difference. Let's choose the first and second terms: 10 and 5.
03

Calculate the Difference

Subtract the second term from the first term: \[ d = 5 - 10 = -5 \]
04

Verify the Common Difference

To ensure the difference is consistent, repeat Step 3 for another pair of consecutive terms. Use the second and third terms: \[ d = 0 - 5 = -5 \] The common difference \(d\) is confirmed as \( -5 \).

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Key Concepts

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

common difference
The common difference is a key concept in arithmetic sequences. It represents the difference between any two consecutive terms in the sequence.

In the given sequence, you can find the common difference by subtracting any term from the one that follows it.
For example, with the terms 10 and 5:
\( d = 5 - 10 \) or \(d = - 5\)

Rechecking with another pair, like 0 and -5, ensures consistency:
\( d = -5 - 0 = -5 \)
This confirms the common difference, which in this case is -5.
sequence
An arithmetic sequence is a list of numbers where each term after the first is found by adding the common difference to the previous term.
A sequence is identified by its starting point and how it progresses.

Take the sample sequence: 10, 5, 0, -5, -10, ...
Each term in this sequence follows a specific order and differs from its predecessor by a constant amount, called the common difference. This pattern allows you to predict the next terms in the sequence easily.
terms
Terms are the individual elements in a sequence.
In our example 10, 5, 0, -5, -10,...
Each number is called a term.

The first term, often denoted as \(a_1\), is crucial in identifying the sequence's start.
Following terms, like \(a_2, a_3\), are derived using the common difference.

Identifying the terms correctly helps in understanding and working with sequences effectively.
For instance, to find the third term if the first term is 10 and the common difference is -5:
\[ a_3 = a_1 + 2d = 10 + 2(-5) = 0\]
arithmetic progression
An arithmetic progression (AP) is another name for an arithmetic sequence.

In an AP, the difference between consecutive terms is always consistent.
This constant difference makes these progressions straightforward to analyze and predict.

An arithmetic progression can be expressed in a general form:
\[a, a + d, a + 2d, a + 3d, \ldots \]
Here, \(a\) is the first term and \(d\) is the common difference.
Using the given sequence again: 10, 5, 0, -5, -10, ..., we see it follows the general form with \(a = 10\) and \(d = -5\).

This format helps in defining the position and value of any term in the progression.

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