Find the Term-to-Term Rule: Sequence 51,47,43,39

What is the term-to-term rule for the sequence below?

$51,47,43,39\ldots$

To determine the term-to-term rule for this sequence, we need to identify the pattern of change between terms. In this sequence, each term is obtained by subtracting 4 from the previous term:

• The first term is 51.
• The second term is 47, which is 51 - 4.
• The third term is 43, which is 47 - 4.
• The fourth term is 39, which is 43 - 4.

Thus, the common difference, $d$, between consecutive terms is $-4$.

We can use the formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n-1) \times d$

Here, $a_1 = 51$ and $d = -4$. Substituting these into the formula gives:

$a_n = 51 + (n-1) \times (-4)$

Expanding this equation, we have:

$a_n = 51 - 4(n-1)$

Simplifying, we get:

$a_n = 51 - 4n + 4$

$a_n = 55 - 4n$

Therefore, the term-to-term rule for this sequence is $a_n = 55 - 4n$.