The given number pattern is 3, 11, 19, 27, extand so on. Write the next three terms. Write a formula for the n^ extth term. Determine the 200^ extth term. Which term has a value of 179?

The given number pattern is $3, 11, 19, 27, \dots$

Step 1: Find the common difference.
The difference between consecutive terms is:
$11 - 3 = 8$
$19 - 11 = 8$
$27 - 19 = 8$
The common difference is $d=8$.

1. Write the next three terms if the pattern continues in the same way.
To find the next terms, add the common difference to the last known term.
The last given term is 27.
Next term: $27 + 8 = 35$
Next term: $35 + 8 = 43$
Next term: $43 + 8 = 51$
The next three terms are $\boxed{35, 43, 51}$.

2. Write a formula for the $n^{th}$ term in the form $T_n = \dots$.
For an arithmetic sequence, the formula for the $n^{th}$ term is $T_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
Here, $a=3$ and $d=8$.
Substitute these values into the formula:

$$T_n = 3 + (n-1)8$$ $$T_n = 3 + 8n - 8$$ $$T_n = 8n - 5$$

The formula for the $n^{th}$ term is $\boxed{T_n = 8n - 5}$.

3. Determine the $200^{th}$ term.
Use the formula $T_n = 8n - 5$ and substitute $n=200$.

$$T_{200} = 8(200) - 5$$ $$T_{200} = 1600 - 5$$ $$T_{200} = 1595$$

The $200^{th}$ term is $\boxed{1595}$.

4. Which term has a value of 179?
Set $T_n = 179$ in the formula $T_n = 8n - 5$ and solve for $n$.

$$179 = 8n - 5$$ $$179 + 5 = 8n$$ $$184 = 8n$$ $$n = \frac{184}{8}$$ $$n = 23$$

The $\boxed{23^{rd}}$ term has a value of 179.