9.2.2B Geometric Sequences & Series

Benchmark Group B

9.2.2.4
Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively.
For example: A closed form formula for the terms t_n in the geometric sequence 3, 6, 12, 24, ... is t_n = 3(2)^n-1, where n = 1, 2, 3, ... , and this sequence can be expressed recursively by writing t₁ = 3 and t_n = 2t_n-1,for n ≥ 2.
Another example: the partial sums s_n of the series 3 + 6 + 12 + 24 + ... can be expressed recursively by writing s₁ = 3 and s_n = 3 + 2s_n-1,for n ≥ 2.

9.2.2.5
Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts.

What students should know and be able to do [at a mastery level] related to these benchmarks

• Students should know that there are infinitely many function types that fit a given sequence of numbers. For example, the sequence 3, 6, ... could be arithmetic (linear), geometric (exponential), or any polynomial function. In fact there are infinitely many quadratic functions that contain the points (1,3) and (2,6). The sequence 3, 6, 12, ... is not arithmetic but could be geometric or quadratic. There is only one quadratic function that would match these three points but there are infinitely many cubic functions that would match. Students should not be led to believe that a sequence of three points like 3, 6, 12 means that the sequence is automatically geometric.
• Students should be able to find a recursive and an explicit formula for a geometric sequence given at least two terms of the sequence and the positions of those terms. Students should know that a recursive rule describes the relationships among the values of the independent variable and that an explicit rule describes a relationship between the value of the independent variable and the value of its corresponding dependent variable.

• Students should know that a recursive rule contains three parts to be considered complete: the rule, a term, and the domain of the indexing values. The rule $t_{n}=2\cdot t_{n-1}$ is incomplete since there are infinitely many sequences that fit this rule (3, 6, 12, 24, ... and 5, 10, 20, 40, ... both fit this rule). A term value could be $t_{4}=24$, but this is not enough since we do not know the domain of the index variable. A complete rule to describe the geometric sequence 3, 6, 12, 24, ... would be $t_{n}=2\cdot t_{n-1}$, $t_{1}=3$, $n\geq 2$. A convention that is often followed in high school is that the value of the indexing variable is an integer.
• Students should know that the domain of a geometric sequence defined by an explicit rule is often restricted to integers. A common convention used is the domain of the function defined explicitly like $y=3\cdot2^{x-1}$ is the set of real numbers unless otherwise noted.
• Students should be able to notice the recursive relationships found when finding the sum of a geometric sequence and be able to write the relationship with a recursive rule.

Students should be able to write the complete rule for this sum recursively as s_n = s_n-1 + t_n, s₁ = 3, n ≥ 2. The rule s_n = s_n-1 + t_n is awkward since it includes two indexing variables. It can be simplified by noticing a pattern in the values of s_n column or a general rule if it is expanded cleverly as is demonstrated in the far right column in the table above. In general if a geometric sequence is defined by t_n = t_n-1∙b, t₁ = a, n ≥ 2, then s_n = a + b∙s_n-1, s₁ = a, n ≥ 2. Students should not memorize this formula but be able to follow the logic behind the following expansion using specific numbers.

• Students should be able to identify situations that can be described by a geometric sequence and make sense of these situations by coming up with a model and using graphing utilities and/or spreadsheets to analyze the model.

Work from previous grades that supports this new learning includes:

• As students explore exponential functions with multiplicative rates of change they have to have a robust understanding of multiplication. They need to move past thinking of multiplication simply as repeated addition (i.e. $3\times20=20+20+20$) and the misconception that multiplication always makes a product that is bigger than either factor. They need to understand that there exists a number that make the equation $\square\times20=15$ true. Students need to understand that to find the product of $\frac{3}{4}\times20$ they need to partition $20$ into four equal amounts of five then iterate the five three times as shown on the number line below.

Thinking of multiplication as both a partitioning and iterating process the students will move beyond thinking of multiplication as repeated addition and they will be able to make sense of functions like $f(x)=20\cdot(\frac{3}{4})^{x}$.

• Students in middle school learn to define arithmetic sequences both recursively and explicitly. They learn the notation for both types of rules and that an arithmetic sequence is a linear function with a restricted domain.