urgent pattern i can not solve

Hello, kl050196!

Mr. Fantastic did his usual excellent job.
I too enjoy cranking out the generating functions for these problems.

However, since it asked for the next term (only),
. . we can use a somewhat intuitive approach.

Quote:

What number comes next in this series: . $3, 10, 21, 44, 87$

. . $(a)\;112\qquad(b)\;125\qquad(c)\;130\qquad(d)\;163 \qquad(e)\;158$

Take the difference of consecutive terms.
Then take the differences of the differences, and so on.

$\begin{array}{cccccccccccc} \text{Sequence} & 3 &&10&&21&&44&&87 \\ \text{1st diff.} & & 7 && 11&&23&&43 \\ \text{2nd diff.} & & & 4 &&12&&20 \\ \text{3rd diff.} & & & & 8 && 8 \end{array}$

It seems that the 3rd differences are constant.

If this is true, we can extend the diagram to the right . . .

We assume that the next 3rd difference is also 8:

. . $\begin{array}{cccccccccccc} 3 &&10&&21&&44&&87 \\ & 7 && 11&&23&&43 \\ & & 4 &&12&&20 \\ & & & 8 && 8 && {\color{red}8}\end{array}$

Then the next 2nd difference must be 28:

. . $\begin{array}{cccccccccccc} 3 &&10&&21&&44&&87 \\ & 7 && 11&&23&&43 \\ & & 4 &&12&&20 && {\color{red}28}\\ & & & 8 && 8 && 8\end{array}$

Then the next 1st difference must be 71:

. . $\begin{array} {cccccccccccc} 3 &&10&&21&&44&&87 \\ & 7 && 11&&23&&43 && {\color{red}71}\\ & & 4 &&12&&20 &&28\\ & & & 8 && 8 && 8\end{array}$

Finally, the next term of the sequence must be 158:

. . $\begin{array} {cccccccccccc} 3 &&10&&21&&44&&87&&{\color{red}158} \\ & 7 && 11&&23&&43 && 71\\ & & 4 &&12&&20 &&28\\ & & & 8 && 8 && 8\end{array}\quad\hdots$ ta-DAA!