Lesson 9 Geometric Meanies Solidify Understanding

Ready

For each set of sequences, find the first five terms. Then compare the growth of the arithmetic sequence and the geometric sequence. Which grows faster? When?

1.

Arithmetic sequence: $f (1) = 3$ , common difference $d = 2$

Geometric sequence: $g (1) = 3$ , common ratio $r = 2$

a.

$n$	Arithmetic $f (n)$	Geometric $g (n)$
$1$
$2$
$3$
$4$
$5$

b.

Which value will be greater, $f (100)$ or $g (100)$ ? Why?

2.

Arithmetic sequence: $f (n) = 4 + 10 (n - 1)$

Geometric sequence: $g (n) = 256 {(\frac{1}{2})}^{n}$

a.

$n$	Arithmetic $f (n)$	Geometric $g (n)$
$1$
$2$
$3$
$4$
$5$

b.

Which value will be greater, $f (100)$ or $g (100)$ ? Why?

3.

Arithmetic sequence: $f (1) = 200, f (n) = f (n - 1) + 10$

Geometric sequence: $g (1) = 4, g (n) = g (n - 1) \cdot 2$

a.

$n$	Arithmetic $f (n)$	Geometric $g (n)$
$1$
$2$
$3$
$4$
$5$

b.

Which value will be greater, $f (100)$ or $g (100)$ ? Why?

4.

Arithmetic sequence: $f (n) = 100 - 10 (n - 1)$

Geometric sequence: $g (n) = 1 {(3)}^{n}$

a.

$n$	Arithmetic $f (n)$	Geometric $g (n)$
$1$
$2$
$3$
$4$
$5$

b.

Which value will be greater, $f (100)$ or $g (100)$ ? Why?

5.

Arithmetic sequence: $f (1) = 128$ , common difference $d = - 2$

Geometric sequence: $g (1) = 128$ , common ratio $r = \frac{1}{2}$

a.

$n$	Arithmetic $f (n)$	Geometric $g (n)$
$1$
$2$
$3$
$4$
$5$

b.

Which value will be greater, $f (100)$ or $g (100)$ ? Why?

6.

Considering arithmetic and geometric sequences, would there ever be a time that a geometric sequence does not outgrow an arithmetic sequence in the long run as the number of terms in the sequences becomes really large? Explain.

Set

Each of the given tables represents a geometric sequence. Find the missing terms in the sequence and create the explicit and recursive equations. Show your method.

7.

a.

Table 1

$x$	$1$	$2$	$3$
$y$	$5$		$20$

b.

Explicit equation:

c.

Recursive equation:

8.

a.

Table 2

$x$	$y$
$1$	$2$
$2$
$3$
$4$	$128$

b.

Explicit equation:

c.

Recursive equation:

9.

a.

Table 3

$x$	$y$
$1$	$3$
$2$
$3$	$108$
$4$

b.

Explicit equation:

c.

Recursive equation:

10.

a.

Table 4

$x$	$y$
$1$	$7$
$2$
$3$
$4$
$5$	$112$

b.

Explicit equation:

c.

Recursive equation:

Go

Use the given information to write the explicit equation for each geometric sequence.

11.

$f (1) = 13$ , common ratio: $r = 4$

12.

$f (1) = 7, f (n) = 1.05 f (n - 1)$

13.

$f (n) = f (n - 1) \cdot 6; f (1) = \frac{7}{3}$

14.

Which of the geometric sequences in problems 11–13 has the greatest value at $f (100)$ ? Why?