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Big Ideas Math Integrated I, 2016

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Big Ideas Math Integrated I, 2016 View details

6. Recursively Defined Sequences

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Exercise 36 Page 319

Make a table to organize the terms. Then graph the ordered pairs.

Graph:

Recursive Rule: $a_{1} = 16, a_{n} = \frac{1}{2} \cdot a_{n - 1}$
Explicit Rule: $a_{n} = 16 (\frac{1}{2})^{n - 1}$

Practice makes perfect

We will find the first four terms of the described sequence, then graph them. After that, we will write a recursive and an explicit rule for the sequence.

Graph

We will make a table to show the first four terms. Note that the first term is $16$ and the common ratio is $\frac{1}{2} .$

Let's plot the ordered pairs $(1, 16),$ $(2, 8),$ $(3, 4),$ and $(4, 2) .$

Recursive Rule

The sequence

a_{n}

is a geometric sequence, with first term

a_{1} = 16

and common ratio

r = \frac{1}{2} .

Now we can write the recursive equation.

a_{n} = r \cdot a_{n - 1} \Rightarrow a_{n} = \frac{1}{2} \cdot a_{n - 1}

The recursive rule is the recursive equation together with the first term.

Recursive Rule : a_{1} = 16, a_{n} = \frac{1}{2} \cdot a_{n - 1}

Explicit Rule

The explicit rule for a geometric sequence is the formula

a_{n} = a_{1} (r)^{n - 1},

where

a_{1}

is the first term and

r

is the common ratio. We know

a_{1} = 16

and

r = \frac{1}{2}

Explicit Rule : a_{n} = 16 (\frac{1}{2})^{n - 1}

Subchapter links

Communicate Your Answer p.313

Monitoring Progress p.315-317

Exercises p.318-320