What is the next number in this sequence 3 -6 12 4 20 ?

Question

What is the next number in this sequence of numbers: 3 -6 12 4 20 ?

Need rule, thank you!

Answer 1

3     -6     12     4      20      13      27     

   -9     18     -8     16     -7      14

 

 

Answer 2

These numbers are related in the following way, beginning with 3: First, we subtract 9 to get -6: 3 - 9= -6. Next, we add 18: -6 + 18= 12.  To get from 12 to 4, we subtract 8: 12 - 8 = 4, and to get from 4 to the final number of 20, we must add 16: 4 + 16= 20.  We need to examine the numbers that we add/take away in the sequence to find a pattern so we can predict the next number.

So, let's look at the relationship between the numbers we added and subtracted to get from 3 to -6, from -6 to 12, and so on.  First, we subtracted 9, then added 18, then subtracted 8, then added 16. Think of ways in which the numbers 9, 18, 8, and 16 are related.  If I start with 9, I have to multiply by 2 to get 18: 9 * 2= 18.  To get from the next number, 8, to the fourth in the sequence, 16, I must multiply by 2 again: 8 * 2=16.  

Now, we know the relationship between 9 and 18, and the relationship between 8 and 16.  We still need to figure out why the pattern moves from adding 18 to subtracting 8.  To do this, I returned to my idea of multiplying by 2, which grouped our numbers as follows: 9 was related to 18, and 8 was related to 16.  We can see that 9 minus 1 equals 8 (9 - 1= 8).  

 So, let's say 9 is "x." The relationship between the numbers in the original sequence can be summed up as follows: First number in the sequence - x, + 2x, - (x-1), + 2(x-1).  Let's plug in 3 as the first number in the sequence and 9 as x to test our pattern: 3 - 9= -6.  -6 + 2(9)= -6 + 18= 12.  12 - (9-1)= 12 - 8= 4. 4 + 2(9-1) = 4+ 2(8) = 4 + 16= 20.  If we put those answers in order, we have 3, -6, 12, 4, 20: the original sequence! 

Using the pattern we found in that equation, we can predict that the next number in the sequence would be related to 20 by subtracting (x-2), where x still equals 9, since in order to get 4 after 12 we had to subtract (x-1) from 12. So, 20 - (x-2) becomes 20 - (9-2) = 20 - 7= 13.  And there's our answer!