The given sequence is $8, 4, 0, 4, \ldots$, and we are asked to find a generalized algebraic form.

Let’s analyze the pattern of the sequence:

• The 1st term is $8$,
• The 2nd term is $4$,
• The 3rd term is $0$,
• The 4th term is $4$.

It appears to oscillate between decreasing and increasing, so let's hypothesize a formula of the form:

$a_n = A \cdot \cos(Bn + C) + D$

Where:

• $A$ determines the amplitude,
• $B$ and $C$ adjust the phase and periodicity,
• $D$ shifts the entire sequence vertically if needed.

1. The sequence seems to have periodicity of 4, as the 5th term is expected to repeat the 1st term. So, we assume $B = \frac{\pi}{2}$ to give a period of 4.
2. The maximum value (at $n = 1$) is 8, suggesting $A = 4$, and the minimum (at $n = 3$) is 0, suggesting no vertical shift (i.e., $D = 4$).

Thus, the formula becomes:

$a_n = 4 \cdot \cos\left(\frac{\pi}{2}(n - 1)\right) + 4$

This represents the oscillating pattern with a period of 4, shifting between 8, 4, 0, and 4.

Would you like further details on how the cosine pattern was chosen? Here are some related questions:

1. Can you explain how periodicity works in sequences?
2. What if the sequence has a different starting point?
3. How do sine and cosine functions model oscillating sequences?
4. What is the role of phase shifts in such formulas?
5. Can we model the sequence using other methods like recursion?

Tip: Understanding cosine functions can help in modeling many periodic or oscillating sequences effectively.