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: a_n = 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 a_n = 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.

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 a_n = a₁r ^n-1
2. Substitute the a₁ and r values only into a_n = 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.

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 a_n = a₁r ^n-1.
• Eq. 2: substitute the other n value into a_n = 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 a_n = a₁r ^n-1.