Question

Select the response that creates an accurate formula for the sequence 2, 3.5, 6.125, 10.71875,...

Answer Options:

y=2+1.75x
y=
1.75x
2
​

y=2(1.75)
x

y=2(1.75)
x−1

Answer 1

To find the correct formula for the sequence \( 2, 3.5, 6.125, 10.71875,... \), let's first analyze the pattern of the sequence.

1. The first term is \( 2 \) which is equal to \( 2 \).
2. The second term is \( 3.5 \), which can be obtained by calculating:
\[
2 + 1.75 = 3.5
\]
3. The third term \( 6.125 \) can be calculated as:
\[
3.5 + 2.625 = 6.125 \quad \text{(where \( 2.625 = 1.75 \times 1.5 \))}
\]
4. The fourth term \( 10.71875 \) can be obtained by calculating:
\[
6.125 + 4.59375 = 10.71875 \quad \text{(where \( 4.59375 \approx 1.75 \times 2.625 \))}
\]

From this pattern, it seems that each term is reliant on the previous term, multiplied by \( 1.75 \) and plus an increasing value.

Now, let's check which of the provided options could form a formula for this sequence.

1. **Option 1: \( y = 2 + 1.75x \)**
- For \( x = 0 \), \( y = 2 + 1.75(0) = 2 \)
- For \( x = 1 \), \( y = 2 + 1.75(1) = 3.75 \) (not a match for \( 3.5 \))
- This option doesn’t fit.

2. **Option 2: \( y = 1.75x + 2 \)**
- Using the same calculations as option 1, this option also yields \( y = 3.75 \) for \( x = 1 \) (not a match for \( 3.5 \)).
- This option doesn’t fit.

3. **Option 3: \( y = 2(1.75)x \)**
- For \( x = 0 \): \( y = 2(1.75)^0 = 2 \) (first term is correct)
- For \( x = 1 \): \( y = 2(1.75)^1 = 3.5 \) (second term is correct)
- For \( x = 2 \): \( y = 2(1.75)^2 = 6.125 \) (third term is correct)
- For \( x = 3 \): \( y = 2(1.75)^3 = 10.71875 \) (fourth term is correct)
- This option fits well.

4. **Option 4: \( y = 2(1.75)x-1 \)**
- Adjusting the function results in incorrect calculations for the sequence.
- This option doesn’t fit.

After evaluating the options, the correct formula for the sequence is:

**\( y = 2(1.75)^x \)** where \( x \) starts at 0 for the first term.

Thus, the suitable response is:
**\( y = 2(1.75)^x \)**.