Question

Is the sequence left-brace a subscript n baseline right-brace equals left-brace 0.4 comma 1.6 comma 6.4 comma 25.6 comma ellipsis right-brace arithmetic, geometric, or neither? If arithmetic, identify the common difference, d. If the sequence is geometric, identify the common ratio, r.

(1 point)
Responses

arithmetic; d = 1.2
arithmetic; d = 1.2

arithmetic; d = 4
arithmetic; d = 4

geometric; r = 1.2
geometric; r = 1.2

geometric; r = 4
geometric; r = 4

neither

Answer 1

To determine whether the sequence \( a_n = \{ 0.4, 1.6, 6.4, 25.6, \ldots \} \) is arithmetic, geometric, or neither, we can analyze the differences between the terms (for arithmetic) and the ratios between consecutive terms (for geometric).

1. **Check for arithmetic:**
- To be an arithmetic sequence, the difference between consecutive terms should be constant.

\[
a_2 - a_1 = 1.6 - 0.4 = 1.2
\]
\[
a_3 - a_2 = 6.4 - 1.6 = 4.8
\]
\[
a_4 - a_3 = 25.6 - 6.4 = 19.2
\]

The differences (1.2, 4.8, 19.2) are not constant, so the sequence is not arithmetic.

2. **Check for geometric:**
- To be a geometric sequence, the ratio between consecutive terms should be constant.

\[
\frac{a_2}{a_1} = \frac{1.6}{0.4} = 4
\]
\[
\frac{a_3}{a_2} = \frac{6.4}{1.6} = 4
\]
\[
\frac{a_4}{a_3} = \frac{25.6}{6.4} = 4
\]

The ratios (4, 4, 4) are indeed constant. Therefore, the sequence is geometric with a common ratio:

\[
r = 4
\]

In conclusion, the correct answer is:

**geometric; r = 4**