Arithmetic Sequence Analysis: Identifying the Term-to-Term Rule

To determine the term-to-term rule for the sequence $4, 5, 6, 7, \ldots$, we should follow these steps:

• Step 1: Identify the difference between consecutive terms.
• Step 2: Establish the rule based on the constant difference.

Let's proceed with the solution:

Step 1: First, notice the differences between consecutive terms in the sequence: $5 - 4 = 1, \quad 6 - 5 = 1, \quad 7 - 6 = 1$

It is clear that each term increases by 1.

Step 2: Since each term increases by 1, the term-to-term rule is to add 1 to the previous term. Therefore, if we denote the $n$-th term by $T_n$, then the subsequent term $T_{n+1}$ can be described by: $T_{n+1} = T_n + 1$

This term-to-term rule can also be expressed in terms of the sequence starting point: $T_n = n + 3$ where $n$ is the term position in the sequence starting from the first term being termed $T_1 = 4$. Hence, choice 1 $(n+3)$ correctly represents each term in the sequence starting from $n = 1$.

Therefore, the term-to-term rule for the sequence is $T_n = n + 3$.