Question 471847
1) 1, 2, 6, 24, ___
2) 0, 3, 8, 15, 24, ___
<pre>
List the numbers this way

n  term
1    1
2    2
3    6
4   24 
5

Now I'll do some coloring:

n term
1   <font color="red">1</font> 
<font color="blue">2</font>   <font color="green">2</font>
3   6
4  24 
5

We notice above that if we multiply the red <font color="red">1</font> by the blue <font color="blue">2</font> we get the green <font color="green">2</font>.

n term
1   1
2   <font color="red">2</font>
<font color="blue">3</font>   <font color="green">6</font>
4  24 
5

We notice above that if we multiply the red <font color="red">2</font> by the blue <font color="blue">3</font> we get the green <font color="green">6</font>.

 
n term
1   1
2   2
3   <font color="red">6</font>
<font color="blue">4</font>  <font color="green">24</font>
5

We notice above that if we multiply the red <font color="red">6</font> by the blue <font color="blue">4</font> we get the green <font color="green">24</font>.


n term
1   1
2   2
3   6
4  <font color="red">24</font> 
<font color="blue">5</font> <font color="green">120</font>

We REASON that in the above if we multiply the red <font color="red">24</font> by the blue <font color="blue">5</font> we must get a green <font color="green">120</font>. 

[This sequence is the sequence of "factorials", indicated by n!, but you don't need
 to know that yet].

-----------------

2) 0, 3, 8, 15, 24, ___

n term
1   0
2   3
3   8
4  15 
5  24
6

Think of another sequence that this one is near.  How about this sequence:

n term
1   1
2   4
3   9
4  16 
5  25
6  36

Each term of our sequence is 1 less than this sequence and this sequence is easily
recognized as the sequence of the squares of n or nē.

So every term of your sequence is 1 less than the corresponding sequence
of squares,

so the next term has to be 36-1 or 35.

Edwin</pre>