1) 1, 2, 6, 24, ___
2) 0, 3, 8, 15, 24, ___
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 1
2 2
3 6
4 24
5
We notice above that if we multiply the red 1 by the blue 2 we get the green 2.
n term
1 1
2 2
3 6
4 24
5
We notice above that if we multiply the red 2 by the blue 3 we get the green 6.
n term
1 1
2 2
3 6
4 24
5
We notice above that if we multiply the red 6 by the blue 4 we get the green 24.
n term
1 1
2 2
3 6
4 24
5 120
We REASON that in the above if we multiply the red 24 by the blue 5 we must get a green 120.
[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