A. Definition
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sequences - an ordered list of numbers.
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terms - the numbers in the sequence.(a variable with a subscript number gives the term place in the sequence such as means the 7th term)
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general term - denoted by an and is the nth term.
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arithmetic sequence - a sequence where the difference “d” between consecutive terms is constant.
- 4, 9, 14, 19, 24, … is an arithmetic sequence because there is a common difference of 5.
- 17, 14, 11, 8, 5, … is an arithmetic sequence because there is a common difference of -3.
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rule - an equation that allows you to find any term in the sequence.
B . The Rule for an Arithmetic Sequence: an = a1 + (n - 1)d
- an is the nth term of the sequence.
- a1 is the first term of the sequence.
- n is the number of terms in the sequence.
- d is the common difference.
- Use only the a1 and d values to write the rule.
C. Writing a Rule When You Are Only Given the Arithmetic Sequence
- Determine the a1 and d values.
- Substitute the a1 and d values into an = a1 + (n - 1)d.
- Simplify the equation.
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D. Writing a Rule When You Know Some Term In the Arithmetic Sequence and the Common Difference.
- Find a1 by substituting the given information into an = a1 + (n - 1)d.
- Substitute the a1 and d values only into an = a1 + (n - 1)d.
- Simplify the equation.
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E. Writing a Rule When You Only Know Two Terms in the Arithmetic Sequence.
- Write a system of equations.
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Eq. 1: substitute the largest n into an = a1 + (n - 1)d.
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Eq. 2: substitute the smallest n into an = a1 + (n - 1)d.
- Simplify each equation.
- Subtract the equations (Eq. 1 - Eq. 2) to find d.
- Substitute the value of d into Eq. 2 (the "smallest equation") to find a1.
- Substitute the values of a1 and d into an = a1 + (n - 1)d.
- Simplify the equation.
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