1. **Problem Statement:** Show that the sequence $51, -1, 1, -1, 1, -1, c, (-1)^{n+1}, c_6$ diverges. 2. **Understanding the sequence:** The sequence appears to start with $51$, then alternates between $-1$ and $1$ repeatedly. The general term from the third term onward seems to be $(-1)^{n+1}$, which alternates between $1$ and $-1$ as $n$ increases. 3. **Definition of convergence:** A sequence $a_n$ converges to a limit $L$ if for every $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $|a_n - L| < \epsilon$. 4. **Check the behavior of the sequence:** From the third term onward, the terms are $1, -1, 1, -1, \ldots$ which oscillate and do not approach a single value. 5. **Conclusion:** Since the terms do not approach a single number but keep oscillating between $1$ and $-1$, the sequence does not converge. 6. **Therefore, the sequence diverges.**