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Algebra II Recipe: Arithmetic Sequences
A. Definition
  1. sequences - an ordered list of numbers.
  2. 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)
  3. general term - denoted by an and is the nth term.
  4. arithmetic sequence - a sequence where the difference “d” between consecutive terms is constant.
  5. 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
  1. an is the nth term of the sequence.
  2. a1 is the first term of the sequence.
  3. n is the number of terms in the sequence.
  4. d is the common difference.
  5. Use only the a1 and d values to write the rule.
C. Writing a Rule When You Are Only Given the Arithmetic Sequence
  1. Determine the a1 and d values.
  2. Substitute the a1 and d values into an = a1 + (n - 1)d.
  3. Simplify the equation.
ExamplesExample:
Write a rule for the nth term of the arithmetic sequence 50, 44, 38, 32, …

D. Writing a Rule When You Know Some Term In the Arithmetic Sequence and the Common Difference.
  1. Find a1 by substituting the given information into an = a1 + (n - 1)d.
  2. Substitute the a1 and d values only into an = a1 + (n - 1)d.
  3. Simplify the equation.
ExamplesExample:
Write a rule for the nth term of an arithmetic sequence with a13 = 30 and a common difference of 3/2.

E. Writing a Rule When You Only Know Two Terms in the Arithmetic Sequence.
  1. Write a system of equations.
    • Eq. 1: substitute the largest n into an = a1 + (n - 1)d.
    • Eq. 2: substitute the smallest n into an = a1 + (n - 1)d.
  2. Simplify each equation.
  3. Subtract the equations (Eq. 1 - Eq. 2) to find d.
  4. Substitute the value of d into Eq. 2 (the "smallest equation") to find a1.
  5. Substitute the values of a1 and d into an = a1 + (n - 1)d.
  6. Simplify the equation.
ExamplesExample:
Write a rule for the nth term of an arithmetic sequence when a6 = 10 and a21 = 55.




G Redden

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