What number comes next in this series????? 3, 10, 21, 44, 87
a)112
b)125
c)130
d)163
e)158
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.
What number comes next in this series: .$\displaystyle 3, 10, 21, 44, 87$
. . $\displaystyle (a)\;112\qquad(b)\;125\qquad(c)\;130\qquad(d)\;163 \qquad(e)\;158$
Take the difference of consecutive terms.
Then take the differences of the differences, and so on.
$\displaystyle \begin{array}{cccccccccccc}
\text{Sequence} & 3 &&10&&21&&44&&87 \\
\text{1st diff.} & & 7 && 11&&23&&43 \\
\text{2nd diff.} & & & 4 &&12&&20 \\
\text{3rd diff.} & & & & 8 && 8 \end{array}$
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:
. . $\displaystyle \begin{array}{cccccccccccc}
3 &&10&&21&&44&&87 \\
& 7 && 11&&23&&43 \\
& & 4 &&12&&20 \\
& & & 8 && 8 && {\color{red}8}\end{array}$
Then the next 2nd difference must be 28:
. . $\displaystyle \begin{array}{cccccccccccc}
3 &&10&&21&&44&&87 \\
& 7 && 11&&23&&43 \\
& & 4 &&12&&20 && {\color{red}28}\\
& & & 8 && 8 && 8\end{array}$
Then the next 1st difference must be 71:
. . $\displaystyle \begin{array}
{cccccccccccc}
3 &&10&&21&&44&&87 \\
& 7 && 11&&23&&43 && {\color{red}71}\\
& & 4 &&12&&20 &&28\\
& & & 8 && 8 && 8\end{array}$
Finally, the next term of the sequence must be 158:
. . $\displaystyle \begin{array}
{cccccccccccc}
3 &&10&&21&&44&&87&&{\color{red}158} \\
& 7 && 11&&23&&43 && 71\\
& & 4 &&12&&20 &&28\\
& & & 8 && 8 && 8\end{array}\quad\hdots$ ta-DAA!