The nth term of a number pattern is as follows. T_n = -5n - 4 if the n is an even number. T_n = -n² + 6 if n is an odd number. Determine the value of: T₆ + T₇ and Determine the value of k if T_k = -219

Since $n=6$ is an even number, we use the formula $T_n = -5n - 4$.

$$T_6 = -5(6) - 4$$ $$T_6 = -30 - 4$$ $$T_6 = -34$$

Since $n=7$ is an odd number, we use the formula $T_n = -n^2 + 6$.

$$T_7 = -(7)^2 + 6$$ $$T_7 = -49 + 6$$ $$T_7 = -43$$

$$T_6 + T_7 = -34 + (-43)$$ $$T_6 + T_7 = -34 - 43$$ $$T_6 + T_7 = -77$$

The value of $T_6 + T_7$ is $\boxed{-77}$.

3.3.2 Determine the value of $k$ if $T_k = -219$

We need to consider two cases for $k$.

If $k$ is even, then $T_k = -5k - 4$.
Set $T_k = -219$:

$$-5k - 4 = -219$$ $$-5k = -219 + 4$$ $$-5k = -215$$ $$k = \frac{-215}{-5}$$ $$k = 43$$

Since we assumed $k$ is even, but 43 is an odd number, this value of $k$ is not valid for this case.

If $k$ is odd, then $T_k = -k^2 + 6$.
Set $T_k = -219$:

$$-k^2 + 6 = -219$$ $$-k^2 = -219 - 6$$ $$-k^2 = -225$$ $$k^2 = 225$$ $$k = \pm\sqrt{225}$$ $$k = \pm 15$$

Since $k$ represents a term number, it must be a positive integer. So, $k=15$.
We assumed $k$ is odd, and 15 is an odd number, so this value of $k$ is valid.

The value of $k$ is $\boxed{15}$.