Determine the Term-to-Term Rule for the Sequence: -1, 3, 7

To solve this problem, we'll determine the term-to-term rule by identifying if this sequence is arithmetic and calculating the common difference.

• Step 1: Calculate the common difference $d$.
  From the given sequence, compute $d = a_2 - a_1 = 3 - (-1) = 4$; similarly, $d = a_3 - a_2 = 7 - 3 = 4$.

• Step 2: Formulate the general term $a_n$ of the sequence.
  Since the sequence has a common difference of $4$, it is an arithmetic sequence. The formula for an arithmetic sequence is given by $a_n = a_1 + (n-1)d$.

• Step 3: Substitute the known values and simplify.
  Using $a_1 = -1$ and $d = 4$, the expression becomes $a_n = -1 + (n-1) \times 4$ which simplifies to $a_n = -1 + 4n - 4 = 4n - 5$.

• Step 4: Verify the formula with the given terms.
  Check $a_1 = 4 \times 1 - 5 = -1$; $a_2 = 4 \times 2 - 5 = 3$; $a_3 = 4 \times 3 - 5 = 7$. All match the given sequence.

Therefore, the term-to-term rule for the sequence is $4n - 5$.

Among the choices provided, the correct option is :

$4n-5$