Find the next three terms in the given sequences and calculate the sum of the arithmetic series.

The sequence is $1, 3, 9, 27, \dots$. Each term is obtained by multiplying the previous term by 3. This is a geometric progression with a common ratio of 3.
$1 \times 3 = 3$
$3 \times 3 = 9$
$9 \times 3 = 27$

The term after 27 is $27 \times 3 = 81$.
The term after 81 is $81 \times 3 = 243$.
The term after 243 is $243 \times 3 = 729$.

The completed sequence is $1, 3, 9, 27, \boxed{81}, \boxed{243}, \boxed{729}$.

The sequence is $1, 3, 6, 10, \dots$. Let's find the differences between consecutive terms:
$3 - 1 = 2$
$6 - 3 = 3$
$10 - 6 = 4$
The differences are increasing by 1 each time ($2, 3, 4, \dots$). This means the next differences will be $5, 6, 7$.

The term after 10 is $10 + 5 = 15$.
The term after 15 is $15 + 6 = 21$.
The term after 21 is $21 + 7 = 28$.

The next three terms are $\boxed{15}, \boxed{21}, \boxed{28}$.

The series is $1 + 3 + 5 + \dots + 39$. This is a sum of consecutive odd numbers.

The $n$-th odd number is given by the formula $2n - 1$. To find the number of terms, we set the last term equal to $2n - 1$:

There are 20 terms in the series.

The sum of the first $n$ odd numbers is given by the formula $n^2$.
Since $n = 20$, the sum is:

The sum is $\boxed{400}$.

The series is $1 + 2 + 3 + 4 + 5 + \dots + 98 + 99 + 100$. This is the sum of the first 100 natural numbers.

The formula for the sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$.
In this case, $n = 100$.

The sum is $\boxed{5050}$.