Arithmetic Sequence Problem: Find the 7th Term in 51, 47, 43, 39, ...

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

• Step 1: Identify the pattern of the sequence.
• Step 2: Use the pattern to find the 7th term.
• Step 3: Verify the solution with the formula for arithmetic sequences.

Now, let's work through each step:

Step 1: Identify the pattern.
The given sequence is $51, 47, 43, 39, \ldots$. Calculating the difference between consecutive terms:

• $47 - 51 = -4$
• $43 - 47 = -4$
• $39 - 43 = -4$

The common difference $d$ is $-4$.

Step 2: Use the pattern to find the 7th term.
We know the sequence is arithmetic with the first term $a_1 = 51$ and common difference $d = -4$. Using the formula for the $n$-th term of an arithmetic sequence:

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

For the 7th term ($n = 7$):

$a_7 = 51 + (7-1) \times (-4) = 51 + 6 \times (-4) = 51 - 24 = 27$

Step 3: Verify.
Confirmed pattern and arithmetic calculation yield the same result.

Therefore, the solution to the problem is $27$.