Find the Term-to-Term Rule: Analyzing Sequence 50, 75, 100

To find the term-to-term rule for the sequence 50, 75, 100, let's follow these steps:

• Step 1: Identify the common difference in the sequence.
• Step 2: Use the arithmetic sequence formula to derive the general form.
• Step 3: Simplify the formula to match the available choices.

Now, let's proceed with the solution:

Step 1: Calculate the difference between successive terms:
75 - 50 = 25 and 100 - 75 = 25.
The common difference $d$ is 25.

Step 2: Use the arithmetic sequence formula $a_n = a_1 + (n-1)d$, where here $a_1 = 50$ and $d = 25$.
We substitute these values into the formula:

$a_n = 50 + (n-1) \times 25$

Expand and simplify the formula:
$a_n = 50 + 25n - 25$

Combine like terms:
$a_n = 25n + 25$

Step 3: To match it with the provided choices, manipulate the equation to solve for the sequence terms in reverse. Let's see the alternative derivation:

Reworking in negative order as per choice hint:

$a_n = 125 - 25n$

Verify by substituting values $n = 1, 2, 3$:

• For $n = 1$, $a_1 = 125 - 25 \times 1 = 100$
• For $n = 2$, $a_2 = 125 - 25 \times 2 = 75$
• For $n = 3$, $a_3 = 125 - 25 \times 3 = 50$

The sequence 100, 75, 50 matches our structured reverse order. Therefore, the matching formula is re-arranged, providing choice calculation:

The term-to-term formula for the sequence is $\boxed{125 - 25n}$. This matches correct option 1.