Arithmetic Sequence Analysis: Is 4 a Term in 51, 47, 43, 39...?

To determine if 4 is part of the sequence $51, 47, 43, 39, \ldots$, we first need to identify the pattern:

• The first term $a_1$ of the sequence is $51$.
• The common difference $d$ is obtained by subtracting successive terms: $47 - 51 = 43 - 47 = 39 - 43 = -4$.

This tells us that the sequence is an arithmetic sequence with a common difference of $-4$.

We express the $n$-th term of the sequence by the formula:

$a_n = a_1 + (n-1) \cdot d = 51 + (n-1)(-4)$

Now, let’s solve the equation to check if $4$ is a term in this sequence:

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

Simplifying,

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

$4 = 51 - 4n + 4$

$4 = 55 - 4n$

Rearranging gives,

$4n = 55 - 4$

$4n = 51$

$n = \frac{51}{4}$

$n = 12.75$

Since $n$ is not a positive integer, 4 is not a term of the sequence.

Therefore, the answer is that the number 4 is not part of the sequence.

The correct answer is: No.