To determine the next term in the sequence [tex]\(1, 4, 10, 19, 31, \ldots\)[/tex], we need to analyze the pattern in the differences between consecutive terms.
Let's begin by listing the given sequence:
[tex]\[ 1, 4, 10, 19, 31 \][/tex]
Next, we find the differences between each pair of consecutive terms:
[tex]\[
\begin{align*}
4 - 1 &= 3 \\
10 - 4 &= 6 \\
19 - 10 &= 9 \\
31 - 19 &= 12 \\
\end{align*}
\][/tex]
So, the differences between consecutive terms are:
[tex]\[ 3, 6, 9, 12 \][/tex]
We observe that the differences themselves follow a pattern. Let's examine the differences between these consecutive differences:
[tex]\[
\begin{align*}
6 - 3 &= 3 \\
9 - 6 &= 3 \\
12 - 9 &= 3 \\
\end{align*}
\][/tex]
The differences between consecutive differences are constant (3). This indicates that the sequence is generated by adding an increasing difference that increases uniformly by 3.
The next difference after 3, 6, 9, 12 can be found by adding 3 to the last difference (12):
[tex]\[
12 + 3 = 15
\][/tex]
Now we add this next difference (15) to the last term in the original sequence (31) to get the next term:
[tex]\[
31 + 15 = 46
\][/tex]
Therefore, the next term in the sequence is:
[tex]\[ \boxed{46} \][/tex]
