The sequence is : 3 8 18 30 70 a b

According to my calculation, a = 112 and b = 192. Here is how I get there. It's pretty sick... English is not my first language so I apologize if some terms are not really appropriate...

3 8 18 30 70
(1x3) (1x8 2x4) (1x18 2x9 3x6) (1x30 2x15 3x10 5x6) (1x70 2x35 5x14 7x10)

I have basically shown here all the multipliers (direct translation from French here...). Now I realised that there were a logic between the common denominators of the numbers. Every number has a common denominator with the following number in the sequence. For instance, 2 is a denominator to both 8 and 18.

It would be possible to know what the missing numbers are (a and b...) if we knew a couple of their multipliers... So we have something here. This is the logical sequence I have come up to :

(1*x3) (1*x8) to (2*x4) (2*x9) to (6*x3) (6*x5) to (5*x6) (5*x14) to (7*x10) (x* y)

(I cannot be sure if (7*x10) is the right way to continue, but I felt strongly it was. It turned out it was.)

This is still the same initial number sequence, but expressed differently. Note that the * is to show the common denominator.

So now I felt strongly that "a" was a multiplier of either 7 or 10. Now I had to find a logical way to find the possible value of the other common denominator. Here is the sequence once more with an added feature :

3 (+5) 8 (+10) 18 (+12) 30 (+40) 70 (+z) a

This is basically the difference between every following number. This is obviously the first thing we looked at when we tried to resolve this sequence lol and it turns out to be important. Obviously, if I could find a value to z, the problem would be solved right away. Earlier, I showed how it was possible to find a logical way to pass from 3 to 8, then to 18 etc. with 1 of the common denominator. This is where we'll use the 2nd denominator ! Once again :

(1*x3) (1*x8) to (2*x4) (2*x9) to (6*x3) (6*x5) to (5*x6) (5*x14) to (7*x10) (x* y)

I'll switch the * to the other number...

(1x3*) (1x8*) to (2x4*) (2x9*) to (6x3*) (6x5*) to (5x6*) (5x14*) to (7x10*) (x y*)

Let's make couples ! (logically...) Since we're gonna be using the difference between every number, jumping from 3 to 8, 8 to 18... We'll try to see if there is some recurence when we jump from 1 number of the sequence to the following with the common denominator... Here is the sequence expressed in the various forms seen so far...


1 2 6 5 7* (common denominators)

(3,8) (4,9) (3,5) (6,14) (10,y) (other multipliers)

(+5) (+10) (+12) (+40) (+z) (differencial)

3 8 18 30 70 a (initial sequence)


Here comes the magic...

1 x (8-3) = 5
2 x (9-4) = 10
6 x (5-3) = 12
5 x (14-6) = 40

Therefore :

7 x (y-10) = z
70 + z = a
a = 7y

Now there are many ways of finding out the answer with these 3 equations... I'll just give you the answer.

a = 112
z = 42
y = 16

And you can use the same logic to find the value of b... Basically, we are now stuck with 112 = (7x16). This is equal to (14x8). The common denominator would be 8. So (8 y). You end up with the following :

b=192
z=80
y=24

AND SO

THE ANSWER TO THE NUMBER SEQUENCE IS :

3 8 18 30 70 112 192.