Verify if 21 Belongs to the Arithmetic Sequence: 51, 47, 43, 39,...

To determine if the number $21$ is part of the given arithmetic sequence $51, 47, 43, 39, \ldots$, we'll follow these steps:

• Step 1: Identify the common difference.
• Step 2: Use the arithmetic sequence formula to check if $21$ is a term.

Step 1: Calculate the common difference, $d$.
The difference between consecutive terms is $51 - 47 = 4$, $47 - 43 = 4$, $43 - 39 = 4$. So, the common difference $d = -4$.

Step 2: Use the formula for the $n$-th term of an arithmetic sequence:

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

Substitute the known values where $a_1 = 51$ and $d = -4$:

$21 = 51 + (n-1)(-4)$

Simplify and solve for $n$:

$21 = 51 - 4n + 4$

$21 = 55 - 4n$

$4n = 55 - 21$

$4n = 34$

$n = \frac{34}{4}$

$n = 8.5$

Since $n$ is not an integer, $21$ is not a term in the sequence.

Therefore, the answer is No.