Question

1,1,2,6,24,_,_,_,_,Explain and fill the sequence, write the explicit and recursive formula for the sequence

Answer 1

Given the sequence:

1, 1, 2, 6, 24

Let's fill in the sequence.

Here, we can see that the nth term is equal to the pprevious term multiplied by n.

First term: 1 x 1 = 1

Second term: 1 x 2 = 2

Third term: 2 x 3 = 6

Fourth term: 6 x 4 = 24

Fifth term: 24 x 5 = 120

Sixth term: 120 x 6 = 720

Seventh term: 720 x 7 = 5040

8th term: 5040 x 8 =40320

Hence, the sequence is:

1, 1, 2, 6, 24, 120, 720, 5040, 40320

The explicit formula of the sequence is:

`a_n=a_(n-1)\ast n`

For example to find the 6th term, we have:

`\begin{gathered} a_6=a_(6-1)\ast6 \\ \\ a_6=a_5\ast6 \\ \\ a_6=120\ast6 \\ \\ a_6=720 \end{gathered}`

We can see the 6th term is 720.

For the recursive formula, we have:

`\begin{cases}a_1=1 \\ \\ n\ge2_{} \\ \\ a_n=a_(n-1)\ast n\end{cases}`

ANSWER:

Sequence: 1, 1, 2, 6, 24, 120, 720, 5040, 40320

Explicit formula:

`a_n=a_(n-1)\ast n`

Recursive formula:

`\begin{cases}a_1=1 \\ \\ n\ge2_{} \\ \\ a_n=a_(n-1)\ast n\end{cases}`