Introduction to Arithmetic Sequences: The Common Difference

Foundational

Learn the fundamental definition of an arithmetic sequence. This stack covers how to identify these sequences by finding their constant 'common difference' (d) and understanding basic sequence notation like a_n.

Introduction to Arithmetic Sequences: The Common Difference

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What is a sequence in mathematics?

A list of numbers arranged in a specific order. Each number in the list is called a term.

What does the notation \(a_n\) represent in a sequence?

It represents the \(n\)-th term in the sequence. For example, \(a_1\) is the first term, and \(a_5\) is the fifth term.

What is the defining characteristic of an arithmetic sequence?

An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant.

What is the 'common difference' of an arithmetic sequence, and what is its symbol?

The common difference is the constant amount added to each term to get the next term. It is denoted by the variable \(d\).

How do you calculate the common difference, \(d\), for an arithmetic sequence?

Subtract any term from its following term: \(d = a_{n+1} - a_n\).

Is the sequence \(5, 8, 11, 14, ...\) arithmetic? If so, what is its common difference?

Yes, it is an arithmetic sequence. The common difference is \(d = 8 - 5 = 3\).

Is the sequence \(20, 15, 10, 5, ...\) arithmetic? If so, what is its common difference?

Yes, it is an arithmetic sequence. The common difference is \(d = 15 - 20 = -5\).

Is the sequence \(2, 4, 8, 16, ...\) an arithmetic sequence?

No. The difference between consecutive terms is not constant (\(4-2=2\), but \(8-4=4\)).

If an arithmetic sequence has \(a_1 = 7\) and a common difference of \(d = 5\), what are the first four terms?

The first four terms are \(7\), \(12\), \(17\), and \(22\).

If an arithmetic sequence has \(a_1 = 10\) and a common difference of \(d = -2.5\), what are the first four terms?

The first four terms are \(10\), \(7.5\), \(5\), and \(2.5\).

The fourth term of an arithmetic sequence is \(19\) and the fifth term is \(23\). What is the common difference, \(d\)?

The common difference is \(d = a_5 - a_4 = 23 - 19 = 4\).

If the terms of an arithmetic sequence are decreasing, what must be true about its common difference, \(d\)?

The common difference \(d\) must be a negative number (\(d < 0\)).

Introduction to Arithmetic Sequences: The Common Difference

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