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# Sequence.

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321
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What are the next 3 terms of this sequence: 1, 2, 3, 8, 15, 48, 105, 384, 945, 3 840..........etc. Thank you very much for help.

Sep 20, 2018
edited by Guest  Sep 20, 2018

#1
+22550
+10

What are the next 3 terms of this sequence:

1, 2, 3, 8, 15, 48, 105, 384, 945, 3 840..........etc.

$$\begin{array}{|rcll|} \hline \mathbf{\text{Sequence:} }&& \mathbf{1,\ 2,\ 3,\ 8,\ 15,\ 48,\ 105,\ 384,\ 945,\ 3840, \ldots} \\ \\ a_1 &=& 1 \\ a_2 &=& 2 \\ a_3 &=& 3 \\ a_4 &=& 8 \\ a_5 &=& 15 \\ a_6 &=& 48 \\ a_7 &=& 105 \\ a_8 &=& 384 \\ a_9 &=& 945 \\ a_{10} &=& 3840 \\ \ldots \\ \mathbf{a_{n} }&\mathbf{=}& \mathbf{n!!} \\ \\ \hline \\ a_{11} &=& 11!! \\ \mathbf{a_{11}} &\mathbf{=}& \mathbf{10395} \\\\ a_{12} &=& 12!! \\ \mathbf{a_{12}} &\mathbf{=}& \mathbf{46080} \\\\ a_{13} &=& 13!! \\ \mathbf{a_{13}} &\mathbf{=}& \mathbf{135135} \\ \hline \end{array}$$

Formula $$n!!$$:

$$\large{ $$n!! = \begin{cases} 2\cdot 4\cdot 6\cdot \ldots \cdot (n-4)\cdot (n-2)\cdot n \text{,} & \text{if } \ n \ \text{ is even} \\ 1\cdot 3\cdot 5\cdot \ldots \cdot (n-4)\cdot (n-2)\cdot n \text{,} & \text{if } \ n \ \text{ is odd} \\ \end{cases}$$ }$$

Example:

$$\begin{array}{|rcll|} \hline 0!! &=& 1 \\ (-1)!! &=& 1 \\ 9!! &=& 1\cdot 3 \cdot 5 \cdot 9 \\ 10!! &=& 2\cdot 4 \cdot 6 \cdot 8 \cdot 10 \\ 19!! &=& 1\cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \cdot 15 \cdot 17 \cdot 19 \\ 20!! &=& 2\cdot 4 \cdot 6 \cdot 8 \cdot 10 \cdot 12 \cdot 14 \cdot 16 \cdot 18 \cdot 20 \\ \hline \end{array}$$

Sep 20, 2018
edited by heureka  Sep 20, 2018
edited by heureka  Sep 21, 2018

#1
+22550
+10

What are the next 3 terms of this sequence:

1, 2, 3, 8, 15, 48, 105, 384, 945, 3 840..........etc.

$$\begin{array}{|rcll|} \hline \mathbf{\text{Sequence:} }&& \mathbf{1,\ 2,\ 3,\ 8,\ 15,\ 48,\ 105,\ 384,\ 945,\ 3840, \ldots} \\ \\ a_1 &=& 1 \\ a_2 &=& 2 \\ a_3 &=& 3 \\ a_4 &=& 8 \\ a_5 &=& 15 \\ a_6 &=& 48 \\ a_7 &=& 105 \\ a_8 &=& 384 \\ a_9 &=& 945 \\ a_{10} &=& 3840 \\ \ldots \\ \mathbf{a_{n} }&\mathbf{=}& \mathbf{n!!} \\ \\ \hline \\ a_{11} &=& 11!! \\ \mathbf{a_{11}} &\mathbf{=}& \mathbf{10395} \\\\ a_{12} &=& 12!! \\ \mathbf{a_{12}} &\mathbf{=}& \mathbf{46080} \\\\ a_{13} &=& 13!! \\ \mathbf{a_{13}} &\mathbf{=}& \mathbf{135135} \\ \hline \end{array}$$

Formula $$n!!$$:

$$\large{ $$n!! = \begin{cases} 2\cdot 4\cdot 6\cdot \ldots \cdot (n-4)\cdot (n-2)\cdot n \text{,} & \text{if } \ n \ \text{ is even} \\ 1\cdot 3\cdot 5\cdot \ldots \cdot (n-4)\cdot (n-2)\cdot n \text{,} & \text{if } \ n \ \text{ is odd} \\ \end{cases}$$ }$$

Example:

$$\begin{array}{|rcll|} \hline 0!! &=& 1 \\ (-1)!! &=& 1 \\ 9!! &=& 1\cdot 3 \cdot 5 \cdot 9 \\ 10!! &=& 2\cdot 4 \cdot 6 \cdot 8 \cdot 10 \\ 19!! &=& 1\cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \cdot 15 \cdot 17 \cdot 19 \\ 20!! &=& 2\cdot 4 \cdot 6 \cdot 8 \cdot 10 \cdot 12 \cdot 14 \cdot 16 \cdot 18 \cdot 20 \\ \hline \end{array}$$

heureka Sep 20, 2018
edited by heureka  Sep 20, 2018
edited by heureka  Sep 21, 2018