Find the next three terms of each arithmetic sequence.

$-5,-2,1,4,\mathrm{..}$

The given arithmetic sequence is: $-5,-2,1,4,\mathrm{...}$

From the given arithmetic sequence, it can be noticed that the first, second, third, and fourth terms of the arithmetic sequence are $-5,-2,1$and 4 respectively.

The common difference of the arithmetic sequence is the difference between the succeeding term and its preceding term.

Therefore, the common difference $\left(d\right)$of the given arithmetic sequence is:

$$\begin{gathered} d=-2-\left(-5\right) \\ =-2+5 \\ =3 \end{gathered}$$

Therefore, the first term and the common difference of the given arithmetic sequence are $-5$and 3 respectively.

The term $\left(a_n\right)$of the given arithmetic sequence is given by:

$a_n=a_1+\left(n-1\right)d$, where $a_1$is the first term and $d$is a common difference.

The next three terms of the given arithmetic sequence are fifth, sixth and seventh terms.

Therefore, the next three terms of the given arithmetic sequence having $a_1=-5$and $d=3$are:

$a_5=a_1+\left(5-1\right)d$

$$\begin{gathered} =-5+\left(4\right)\left(3\right) \\ =-5+12 \\ =7 \end{gathered}$$

$a_6=a_1+\left(6-1\right)d$

$$\begin{gathered} =-5+\left(5\right)\left(3\right) \\ =-5+15 \\ =10 \end{gathered}$$

$a_7=a_1+\left(7-1\right)d$

$$\begin{gathered} =-5+\left(6\right)\left(3\right) \\ =-5+18 \\ =13 \end{gathered}$$

Therefore, the next three terms of the given arithmetic sequence are 7, 10 and 13.