Question

Which equation represents the sequence below?

400, 100, 25, . . .

(1 point)
Responses

y=0.75x+1600y is equal to 0 point 7 5 x plus 1600 - no response given

y=0.25x+1600y is equal to 0 point 2 5 x plus 1600 - no response given

y=1600(0.25)xy is equal to 1600 times 0 point 2 5 to the x th power - no response given

y=1600(0.75)x

Answer 1

To find the equation that represents the sequence \(400, 100, 25, \ldots\), we first note the pattern in which the numbers are decreasing.

1. **Identify the pattern**:
- From \(400\) to \(100\), the value is multiplied by \(0.25\) (or divided by \(4\)).
- From \(100\) to \(25\), the value is again multiplied by \(0.25\) (or divided by \(4\)).

This suggests that each term can be derived from the previous term by multiplying by \(0.25\).

2. **Identify the first term**:
- The first term \(y(0)\) can be seen as \(400\).

3. **General formula**:
The \(n\)-th term of the sequence can be represented as:
\[
y(n) = 400 \cdot (0.25)^n
\]

4. **Expressing it with a similar format**:
We can express \(400\) as \(1600 \cdot 0.25\) because:
\[
400 = 1600 \cdot 0.25^1
\]
Thus, the equation for the sequence can be represented as:
\[
y = 1600 (0.25)^n
\]

Given the options presented, the equation that matches our derived formula is:

\[ y = 1600(0.25)^x \]

So the correct response is:
**y=1600(0.25)^x**.