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

Video Solution

Solution Steps

00:00 Find the sequence formula

00:04 Identify the first term according to the given data

00:09 Notice the constant difference between terms

00:14 This is the constant difference

00:18 Use the formula to describe an arithmetic sequence

00:22 Substitute appropriate values and solve to find the sequence formula

00:39 Open parentheses properly, multiply by each factor

00:43 Continue solving

00:54 And this is the solution to the question

Step-by-Step Solution

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 :

$4n-5$