Arithmetic Sequence Check: Does 15 Appear in 51,47,43,39...?

To solve this problem, we'll follow these steps:

• Step 1: Determine the common difference of the sequence.
• Step 2: Use the arithmetic sequence formula.
• Step 3: Solve for $n$ and check its validity.

Now, let's work through each step:

Step 1:
The first term of the sequence is $a_1 = 51$.
The second term is $47$, so the common difference $d = 47 - 51 = -4$.

Step 2:
Use the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n-1) \cdot d$.
For $a_n = 15$, we have:
$15 = 51 + (n-1) \cdot (-4)$

Step 3: Solve for $n$:
$15 = 51 - 4n + 4$
$15 = 55 - 4n$
$4n = 55 - 15$
$4n = 40$
$n = \frac{40}{4}$
$n = 10$

Since $n = 10$ is a positive integer, this means that $15$ is the 10th term in the sequence.

Therefore, the number $15$ does appear in the sequence.