Four students wrote sequences in math class.

Angela: -6, -9, -12, -15, ...

Bradley: -2, -6, -12, -24, ...

Carter: -1, -3, -9, -27, ...

Dominique: -1, -3, -9, -81, ...

Which student wrote a geometric sequence?

A. Angela
B. Bradley
C. Carter
D. Dominique



Answer :

To determine which student's sequence is geometric, we need to examine the ratio between consecutive terms in each sequence. A sequence is geometric if the ratio between consecutive terms is constant.

### Angela's Sequence: [tex]\(-6, -9, -12, -15, \ldots\)[/tex]

Let's find the ratio between the consecutive terms:
[tex]\[ \frac{-9}{-6} = \frac{-3}{-2} \][/tex]
[tex]\[ \frac{-12}{-9} = \frac{-4}{-3} \][/tex]
[tex]\[ \frac{-15}{-12} = \frac{-5}{-4} \][/tex]

The ratios are not equal ([tex]\(\frac{3}{2} \neq \frac{4}{3} \neq \frac{5}{4}\)[/tex]), so Angela's sequence is not geometric.

### Bradley's Sequence: [tex]\(-2, -6, -12, -24, \ldots\)[/tex]

Let's find the ratio between the consecutive terms:
[tex]\[ \frac{-6}{-2} = 3.0 \][/tex]
[tex]\[ \frac{-12}{-6} = 2.0 \][/tex]
[tex]\[ \frac{-24}{-12} = 2.0 \][/tex]

The ratios are not equal ([tex]\(3.0 \neq 2.0\)[/tex]), so Bradley's sequence is not geometric.

### Carter's Sequence: [tex]\(-1, -3, -9, -27, \ldots\)[/tex]

Let's find the ratio between the consecutive terms:
[tex]\[ \frac{-3}{-1} = 3.0 \][/tex]
[tex]\[ \frac{-9}{-3} = 3.0 \][/tex]
[tex]\[ \frac{-27}{-9} = 3.0 \][/tex]

The ratios are equal ([tex]\(3.0 = 3.0 = 3.0\)[/tex]), so Carter's sequence is geometric.

### Dominique's Sequence: [tex]\(-1, -3, -9, -81, \ldots\)[/tex]

Let's find the ratio between the consecutive terms:
[tex]\[ \frac{-3}{-1} = 3.0 \][/tex]
[tex]\[ \frac{-9}{-3} = 3.0 \][/tex]
[tex]\[ \frac{-81}{-9} = 9.0 \][/tex]

The ratios are not equal ([tex]\(3.0 \neq 9.0\)[/tex]), so Dominique's sequence is not geometric.

### Conclusion
Based on the analysis above, only Carter's sequence is geometric with a constant ratio of [tex]\(3.0\)[/tex].