Question

Select the response that creates an accurate formula for the sequence 2, 3.5, 6.125. 10.71875 . . .
.(1 point)
Responses

y=21.5x
y equals Start Fraction 2 over 1.5 x End Fraction

y=2+1.75x
y equals 2 plus 1.75 x

y=2(1.75)x−1
y equals 2 left parenthesis 1.75 right parenthesis superscript x baseline

y=2(1.5)x

Answer 1

To find the correct formula for the sequence \( 2, 3.5, 6.125, 10.71875, \ldots \), we need to identify a pattern or a relationship.

Let's analyze the provided sequence:

1. The first term \( a_1 = 2 \).
2. The second term \( a_2 = 3.5 \).
3. The third term \( a_3 = 6.125 \).
4. The fourth term \( a_4 = 10.71875 \).

To find a connection, we can look at the ratios of consecutive terms:

- \( \frac{a_2}{a_1} = \frac{3.5}{2} = 1.75 \)
- \( \frac{a_3}{a_2} = \frac{6.125}{3.5} = 1.75 \)
- \( \frac{a_4}{a_3} = \frac{10.71875}{6.125} = 1.75 \)

The ratio between consecutive terms appears to be consistently \( 1.75 \).

This suggests a geometric sequence where the first term is \( 2 \) and the common ratio is \( 1.75 \).

The general formula for a geometric sequence can be written as:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_1 = 2 \)
- \( r = 1.75 \)

Hence, the formula for the \( n \)-th term would be:

\[
y = 2 \cdot (1.75)^{(n-1)}
\]

Looking at the provided response options, this corresponds to:

**y = 2(1.75)^{x-1}**

This is indeed the correct formula for the given sequence.