What number comes next in this series????? 3, 10, 21, 44, 87

a)112

b)125

c)130

d)163

e)158

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- Mar 12th 2009, 12:53 AMkl050196urgent pattern i can not solve
What number comes next in this series????? 3, 10, 21, 44, 87

a)112

b)125

c)130

d)163

e)158 - Mar 12th 2009, 02:20 AMADARSH
- Mar 12th 2009, 02:28 AMkl050196
- Mar 12th 2009, 03:05 AMmr fantastic
- Mar 12th 2009, 05:57 AMSoroban
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: .$\displaystyle 3, 10, 21, 44, 87$

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

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!*