Arithmetic Sequence Analysis: Does 28 Belong in 51,47,43,39...?

To solve this problem, follow these steps:

• Step 1: Identify the given information in the arithmetic sequence.
• Step 2: Set up the formula for the nth term of an arithmetic sequence.
• Step 3: Solve for $n$ and check if it's a natural number.

Now, let's work through each step:
Step 1: The first term $a_1 = 51$, and the common difference $d = -4$.

Step 2: The formula for the nth term $a_n$ of an arithmetic sequence is:

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

Substituting the known values to find if 28 is an element of this series:

$28 = 51 + (n-1) \cdot (-4)$

Step 3: Simplify and solve for $n$:

$28 = 51 - 4(n-1)$

$28 = 51 - 4n + 4$

$28 = 55 - 4n$

$4n = 55 - 28$

$4n = 27$

$n = \frac{27}{4} = 6.75$

Since $n$ must be a natural number (a positive integer) to indicate a position in the sequence, and 6.75 is not a natural number, we conclude that 28 is not part of the sequence.

Therefore, the solution to the problem is No.