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Algebra II Recipe: Geometric Sequences
A. Definitions
  1. geometric sequence - a sequence where the ration of any term to the previous term is constant.
  2. common ratio -  the constant ratio that is denoted by r.
B. The Rule for a Geometric Sequence: an = a1r n-1
  1. a 1 is the first term.
  2. r is the common ratio.
  3. Use only the a1 and r values to write the rule.
C. Writing a Fule When You Are Only Given the Geometric Sequence.
  1. Determine the a1 and r values.
  2. Substitute the a1 and r values into an = a1r n-1.
  3. The a1 and r values can't be multiplied because r is the base of n-1, it's not the product of a1 and r.
ExamplesExample:
Write a rule for the nth term of the geometric sequence -8, -12, -18, -27, … then find a8.

D. Writing a Rule When You Know Some Term in the Geometric Sequence and the Common Ratio.
  1. Find a1 by substituting the given information into an = a1r n-1
  2. Substitute the a1 and r values only into an = a1r n-1
  3. The a1 and r values can't be multiplied because r is the base of n-1, it's not the product of a1 and r.
ExamplesExample:
Write a rule for the nth term of a geometric sequence with a3 = 5 and a common ratio of 2.

E. Writing a Rule When You Only Know Two Terms in the Geometric Sequence.
  1. Write a system of equations.
    • Eq. 1: substitute one of the n values into an = a1r n-1.
    • Eq. 2: substitute the other n value into an = a1r n-1.
  2. Simplify each equation.
  3. Solve one of the equations for a1.
  4. Substitute this expression for a1 into the other equation to find r. (Solving systems by substituting.)
  5. Substitute the r value into either Eq. 1 or Eq. 2 to find a1.
  6. Substitute the a1 and r values into an = a1r n-1.
ExamplesExample:
Write a rule for the nth term of a geometric sequence when a2 = 45 and a5 = -1215




G Redden

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