Geometric+Sequences+and+Series


 * // Geometric Sequences and Series //**


 * Standard & Assessment Anchor || 2.8.11.C Use patterns, sequences and series to solve routine and non-routine problems. ||
 * Objective || Students will find the indicated terms of a geometric sequence and find the sums of geometric series. ||

In a **// geometric sequence //**, the ratio of successive terms is a constant called the //**common ratio r**// (r ≠ 1) Remember that exponential functions have a common ratio. When we graph the ordered pairs (n, an) of a geometric sequence, the points lie on an exponential curve. We can think of a geometric sequence as an exponential function with sequential natural numbers as the domain. If we examine the sequence 5, 1, 0.2, 0.04, ... we see that the differences are -4, - 0.8 and -0.16, so this is not an arithmetic sequence. The ratios of the terms are 0.2, 0.2, 0.2, so this is could be a geometric sequence with r = 0.2. Each term in a geometric sequence is the product of the previous term and the common ratio, so the recursive rule for a geometric sequence is We can also use an explicit rule to find the nth term of a geometric sequence. Each term is the product of the first term and a power of the common ratio. We can also find a given term in a geometric sequence when we are given two terms; in this case, we must be sure to consider both positive and negative values for r.

//** Geometric means **// are the terms between any two nonconsecutive terms of a geometric series. //If a and b are positive terms of a geometric sequence with exactly one term between them, the geometric mean is given by the expression √ab// For example, the geometric mean of 16 and 25 is //√(16)(25) = √400 or 20.// The indicated sum of the terms of a geometric sequence is called a **// geometric series //**. We can derive a formula for the partial sum of a geometric series by subtracting the product of Sn and r from Sn. Here is a chart showing the formula for the partial sum:
 * // Geometric Means and Series //**

We work with geometric series the same way that we worked with arithmetic series, except we use the appropriate formulas for the geometric series. The method for finding terms of a geometric sequence and the sum of an indicated number of terms of a geometric series on the TI 83+ calculator are the same as for the arithmetic sequences and series, so they will not be repeated here.