Find any term of a geometric sequence, given the formula of the sequence

The  exercise appears under the Precalculus Math Mission. This exercise experiments with recursive and explicit forms of geometric sequences.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find the appropriate term recursively: This problem provides a recursive rule for a geometric sequence. The student is expected to use the rule to find a specified term in the sequence.Geoseq2-1.png
 * 2) Find the appropriate term in the explicit formula: This problem provides an explicit rule for a geometric sequence. The student is expected to use the rule to find a specified term in the sequence.Geoseq2-2.png

Strategies
Knowledge of formulas and patterns from geometric sequences and series are encouraged to ensure success on this exercise.
 * 1) For a recursive formula, the general strategy involves finding the first term, plugging that in to get the second, plugging that in to get the third, et cetera.
 * 2) Generally, the nth term of a geometric sequence is given by $$ g_n=g_1r^{n-1}$$ where g_1 is the first term and r is ratio between successive terms.

Real-life Applications

 * 1) Sequences are series are fundamental to the calculus, which has many applications to real life.
 * 2) Sequences are common on IQ tests for measuring pattern recognition ability.
 * 3) Taylor Series can be used to approximate complicated functions via polynomials.
 * 4) Geometric sequences appear in the study of fractal geometry.
 * 5) Physics has applications for geometric sequences when measuring motion, for example, the  distance travelled by bouncing balls.