Questions: Matching 20 points Match each geometric sequence with corresponding nth term. Geometric Sequences A. 1, 3, 9, 27, ... B. 1, -2, 4, -8, ... nth Term I. (2/3)^n C. 1/2, 1/4, 1/8, 1/16, ... II. (3)^(n-1) D. 2/3, 4/9, 8/27, 16/81, ... III. (-2)^(n-1) IV. 1/2^n c. A. B. D.

Matching 20 points
Match each geometric sequence with corresponding nth term.
Geometric Sequences
A. 1, 3, 9, 27, ...
B. 1, -2, 4, -8, ...
nth Term
I. (2/3)^n
C. 1/2, 1/4, 1/8, 1/16, ...
II. (3)^(n-1)
D. 2/3, 4/9, 8/27, 16/81, ...
III. (-2)^(n-1)
IV. 1/2^n
c.
A.
B.
D.

Transcript text: 4 Matching 20 points Match each geometric sequence with orresponding $n^{\text {th }}$ term. Geometric Sequences A. $1,3,9,27, \ldots$ B. $1,-2,4,-8, \ldots$ $n^{\text {th }}$ Term I. $\left(\frac{2}{3}\right)^{n}$ C. $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$ II. $(3)^{(n-1)}$ D. $\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots$ III. $(-2)^{(n-1)}$ IV. $\frac{1}{2^{n}}$ c. $\qquad$ A. $\qquad$ $\square$ $\square$ B. $\qquad$ $\square$ D. $\qquad$

Solution

Solution Steps

To match each geometric sequence with its corresponding $ n^{\text{th}} $ term, we need to identify the common ratio for each sequence and then match it with the given formulas.

Sequence A: $ 1, 3, 9, 27, \ldots $
- Common ratio: $ 3 $
- $ n^{\text{th}} $ term: $ 3^{(n-1)} $
Sequence B: $ 1, -2, 4, -8, \ldots $
- Common ratio: $ -2 $
- $ n^{\text{th}} $ term: $ (-2)^{(n-1)} $
Sequence C: $ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots $
- Common ratio: $ \frac{1}{2} $
- $ n^{\text{th}} $ term: $ \frac{1}{2^n} $
Sequence D: $ \frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots $
- Common ratio: $ \frac{2}{3} $
- $ n^{\text{th}} $ term: $ \left(\frac{2}{3}\right)^n $

Solution Approach

Identify the common ratio for each sequence.
Match the common ratio with the given $ n^{\text{th}} $ term formulas.

Step 1: Identify the Sequences

We have the following geometric sequences:

$ A: 1, 3, 9, 27, \ldots $
$ B: 1, -2, 4, -8, \ldots $
$ C: \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots $
$ D: \frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots $

Step 2: Determine the Common Ratios

For each sequence, we calculate the common ratio:

For sequence $ A $, the common ratio $ r_A = \frac{3}{1} = 3 $.
For sequence $ B $, the common ratio $ r_B = \frac{-2}{1} = -2 $.
For sequence $ C $, the common ratio $ r_C = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} $.
For sequence $ D $, the common ratio $ r_D = \frac{\frac{4}{9}}{\frac{2}{3}} = \frac{2}{3} $.

Step 3: Match with $ n^{\text{th}} $ Terms

Now we match each sequence with its corresponding $ n^{\text{th}} $ term:

Sequence $ A $ corresponds to $ n^{\text{th}} $ term $ II: 3^{(n-1)} $.
Sequence $ B $ corresponds to $ n^{\text{th}} $ term $ III: (-2)^{(n-1)} $.
Sequence $ C $ corresponds to $ n^{\text{th}} $ term $ IV: \frac{1}{2^n} $.
Sequence $ D $ corresponds to $ n^{\text{th}} $ term $ I: \left(\frac{2}{3}\right)^n $.