The question states that the sequence is arithmetic, which means we find the next number in the sequence by adding (or subtracting) a constant term. We know two of the values, separated by one unknown value.

We know that $\displaystyle n$ is equally far from -1 and from 13; therefore $\displaystyle n$ is equal to half the distance between these two values. The distance between them can be found by adding the absolute values.

$\displaystyle \frac{(\left |13 \right |+\left | -1\right |)}{2}=arithmetic \ constant$

$\displaystyle \frac{14}{2}=7$

The constant in the sequence is 7. From there we can go forward or backward to find out that $\displaystyle n=6$.

$\displaystyle -1+7=6$

$\displaystyle 13-7=6$

By taking the difference between two adjacent numbers in the sequence, we can see that the common difference increases by one each time.

$\displaystyle 3-1=2$

$\displaystyle 6-3=3$

$\displaystyle 10-6=4$

$\displaystyle 15-10=5$

$\displaystyle 21-15=6$

Our next term will fit the equation $\displaystyle x-21=7$, meaning that the next term must be $\displaystyle 28$.

After $\displaystyle 28$, the next term will be $\displaystyle y-28=8$, meaning that the next term must be $\displaystyle 36$.

Finally, after $\displaystyle 36$, the next term will be $\displaystyle z-36=9$, meaning that the next term must be $\displaystyle 45$

The question asks for the sum of the next three terms, so now we need to add them together.

$\displaystyle 28 + 36 + 45 = 109$

First, find a pattern in the sequence.  You will notice that each time you move from one number to the very next one, it increases by 7.  That is, the difference between one number and the next is 7.  Therefore, we can add 7 to 36 and the result will be 43.  Thus $\displaystyle x=43$.

The sequence follows the pattern for the equation:

$\displaystyle n_x=2n_{x-1}+3$

$\displaystyle n_{x-1}=77$

Therefore,

$\displaystyle n_x=2(77)+3=154+3=157$

Determine what kind of sequence you have, i.e. whether the sequence changes by a constant difference or a constant ratio. You can test this by looking at pairs of numbers, but this sequence has a constant difference (arithmetic sequence).

$\displaystyle -27-(-11)=-16$

$\displaystyle -43-(-27)=-16$

So the sequence advances by subtracting 16 each time. Apply this to the last given term.

$\displaystyle -59-(-16)=75$

First, find the common difference for the sequence. Subtract the first term from the second term.

$\displaystyle 14-10=4$

Subtract the second term from the third term.

$\displaystyle 18-14=4$

Subtract the third term from the fourth term.

$\displaystyle 22-18=4$

To find the next value, add $\displaystyle 4$ to the last given number.

$\displaystyle 22+4=26$

First, find the common difference for the sequence. Subtract the first term from the second term.

$\displaystyle 22-15=7$

Subtract the second term from the third term.

$\displaystyle 29-22=7$

To find the next value, add $\displaystyle 7$ to the last given number.

$\displaystyle 29+7=36$

First, find the common difference for the sequence. Subtract the first term from the second term.

$\displaystyle -10-(-2)=-8$

Subtract the second term from the third term.

$\displaystyle -18-(-10)=-8$

To find the next value, add $\displaystyle -8$ to the last given number.

$\displaystyle -18+(-8)=-26$

First, find the common difference for the sequence. Subtract the first term from the second term.

$\displaystyle 19-10=9$

Subtract the second term from the third term.

$\displaystyle 28-19=9$

To find the next value, add $\displaystyle 9$ to the last given number.

$\displaystyle 28+9=37$

First, find the common difference for the sequence. Subtract the first term from the second term.

$\displaystyle -8-(-15)=7$

Subtract the second term from the third term.

$\displaystyle -1-(-8)=7$

Subtract the third term from the fourth term.

$\displaystyle 6-(-1)=7$

To find the next value, add $\displaystyle 7$ to the last given number.

$\displaystyle 6+7=13$