Arithmetic Sequence Analysis: Is 7 Part of 51, 47, 43, 39,...?

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

• Step 1: Identify the pattern and common difference ($d$)
• Step 2: Write the general nth term formula for the sequence
• Step 3: Substitute $a_n = 7$ into the equation and solve for $n$
• Step 4: Check if $n$ is a positive integer

Now, let's work through each step:

Step 1: We observe that the sequence $51, 47, 43, 39, \ldots$ is decreasing by $4$ each time. Thus, the common difference $d = -4$.

Step 2: The first term $a_1 = 51$. The nth term of the sequence can be expressed as:
$a_n = a_1 + (n-1)d$
This simplifies to:
$a_n = 51 + (n-1)(-4)$

Step 3: Set $a_n = 7$ and solve for $n$:
$7 = 51 + (n-1)(-4)$
$7 = 51 - 4(n-1)$
$7 = 51 - 4n + 4$
$7 = 55 - 4n$
$-48 = -4n$
$n = \frac{48}{4} = 12$

Step 4: Since $n = 12$ is a positive integer, $7$ is indeed part of the sequence as the 12th term.

Therefore, the number $7$ is part of the sequence.

Yes