introduction arithmetic sequences are linear functions that have a domain of positive consecutive...

IntroductionArithmetic sequences are linear functions that have a domain of positive consecutive integers in which the difference between any two consecutive terms is equal. Arithmetic sequences can be represented by formulas, either explicit or recursive, and those formulas can be used to find a certain term of the sequence or the number of a certain value in the sequence. An explicit formula is a formula used to find the nth term of a sequence and a recursive formula is a formula used to find the next term of a sequence when the previous term is known.

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3.8.1: Arithmetic Sequences

Key Concepts• An arithmetic sequence is a list of terms separated by

a common difference, the number added to each consecutive term in an arithmetic sequence.

• An arithmetic sequence is a linear function with a domain of positive consecutive integers in which the difference between any two consecutive terms is equal.

• The rule for an arithmetic sequence can be expressed either explicitly or recursively.

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Key Concepts, continued• The explicit rule for an arithmetic sequence is

an = a1 + (n – 1)d, where a1 is the first term in the sequence, n is the term, d is the common difference, and an is the nth term in the sequence.

• The recursive rule for an arithmetic sequence is an = an – 1 + d, where an is the nth term in the sequence, an – 1 is the previous term, and d is the common difference.

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Common Errors/Misconceptions• identifying a non–arithmetic sequence as arithmetic

• defining the common difference, d, in a decreasing sequence as a positive number

• incorrectly using the distributive property when finding the nth term with the explicit formula

• forgetting to identify the first term when defining an arithmetic sequence recursively

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Guided Practice

Example 2Write a linear function that corresponds to the following arithmetic sequence.

8, 1, –6, –13, …

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Guided Practice: Example 2, continued

1. Find the common difference by subtracting two successive terms.

1 – 8 = –7

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2. Confirm that the difference is the same between all of the terms.

–6 – 1 = –7 and –13 – (–6) = –7

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3. Identify the first term (a1).

a1 = 8

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4. Write the explicit formula.an = a1 + (n – 1)d Explicit formula for any given

arithmetic sequence

an = 8 + (n – 1)(–7) Substitute values for a1 and d.

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5. Simplify the explicit formula.an = 8 – 7n + 7 Distribute –7 over (n – 1).

an = –7n + 15 Combine like terms.

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6. Write the formula in function notation.ƒ(x) = –7x + 15

Note that the domain of an arithmetic sequence is positive consecutive integers.

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http://walch.com/ei/CAU3L8S1ArithSeqExplfromVal

Guided Practice

Example 3An arithmetic sequence is defined recursively by an = an – 1 + 5, with a1 = 29. Find the first 5 terms of the sequence, write an explicit formula to represent the sequence, and find the 15th term.

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1. Using the recursive formula: a1 = 29

a2 = a1 + 5

a2 = 29 + 5 = 34

a3 = 34 + 5 = 39

a4 = 39 + 5 = 44

a5 = 44 + 5 = 49

The first five terms of the sequence are 29, 34, 39, 44, and 49.

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2. The first term is a1 = 29 and the common difference is d = 5, so the explicit formula is an = 29 + (n – 1)5.

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3. Simplify.an = 29 + 5n – 5

an = 5n + 24 Combine like terms.

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4. Substitute 15 in for n to find the 15th term in the sequence.

a15 = 5(15) + 24

a15 = 75 + 24

a15 = 99

The 15th term in the sequence is 99.

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http://walch.com/ei/CAU3L8S1ArithSeqExplfromRec

introduction arithmetic sequences are linear functions that have a domain of positive consecutive...

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