3 -6 12 4 20
List them in a column
3
-6
12
4
20
Subtract each number from the one just below it
and write the difference out besides it.
3 -9
-6 18
12 -8
4 16
20
The second column goes -9,18,-8,16
That has this pattern: -9,18,-8,16,-7,14,-6,12,-5,10,-4,8,...
So put -7 below 16 and add 20, get 13.
3 -9
-6 18
12 -8
4 16
20 -7
13
---------------------------
Next number in this pattern
Do the same as the preceding sequence:
7 9
16 -8
8 19
27 -18
9
That has this pattern: 9,-8,19,-18,29,-28,39,-38,48,-47,...
The second column goes 9,-8,19,-18
That has this pattern: 9,-8,19,-18,29,-28,39,-38,48,-47,...
So put 29 below -18 and add 9, get 38.
7 9
16 -8
8 19
27 -18
9 29
38
--------------
Try that method with the next one:
2 5
7 21
26 75
101 299
400
That doesn't seem to work here. Let's try dividing
instead of subtracting
2 3.5
7 3.7142...
26 3.8846...
101 3.96049...
400
Those all round up to 4, and they're getting closer
and closer to being like a geometric sequence with
common ratio 4, so let's see what happens when we
multiply them all by 4.
2 8
7 28
26 104
101 404
400
From 8 we must subtract 1 to get 7,
From 28 we must subtract 2 to get 26,
From 104 we must subtract 3 to get 101,
From 404 we must subtract 4 to get 400,
2 8-1
7 28-2
26 104-3
101 404-4
400
So to get the next number, we multiply 400 by 4 and subtract 5,
2 8-1
7 28-2
26 104-3
101 404-4
400 1600-5
1595
-------------------------------------
7 21 8 72 9
We see a pattern with 1st, 3rd, and 5th
terms, 7, 8, and 9, and we notice that 21
is 7x3, and that 72 is 8x9. And 9 is 32.
So the 5th term 9 must be multiplied by
the next power of 3, which is 33 or 27.
So the next term is 9x33 or 9x27 or 243.
Edwin
