Solve 2 3 4 15 12 3 4 5 28 20 4 5 6 45 30 5 6 7 66 42 6 7 8 ? 56

. .2 . . . . 3 . . . . 4 . . . .15 . . . . 12
. .3 . . . . 4 . . . . 5 . . . .28 . . . . 20
. .4 . . . . 5 . . . . 6 . . . .45 . . . . 30
. .5 . . . . 6 . . . . 7 . . . .66 . . . . 42
. .6 . . . . 7 . . . . 8 . . . .?? . . . . 56
(n+1). .(n+2) . .(n+3). . . |. . . . .(n+2) (n+3)
. . . . . . . . . . . . . . . . . . . |
. . . . . . . . . . . . . . (n+2) (2n+3)

Missing number : (5 + 2) (10 + 3) = 91

91

The list of numbers appears to be 5 interleaved series. They seem to be
Series 1: {2, 3, 4, 5, 6}  a linear series with a difference of 1
Series 2: {3, 4, 5, 6, 7}  a linear series with a difference of 1
Series 3: {4, 5, 6, 7, 8}  a linear series with a difference of 1
Series 4: {15, 28, 45, 66, ?}  a quadratic series, with difference increasing by 4 at each step. The difference between the missing number and 66 is 25, so the missing number is 91.
Series 5: {12, 20, 30, 42, 56}  a quadratic series, with the difference increasing by 2 at each step.

2 3 4 15 12
3 4 5 28 20
4 5 6 45 30
5 6 7 66 42
6 7 8 91 56
is 91
**1/  4+2=6, 6*2=12 , 12+3=15, 3*4= 12 ( 2 3 4 15 12)
**2/ 5+3= 8, 8*3 = 24, 24+4= 28, 4*5= 20 ( 3 4 5 28 20)
**3/ 6+4 = 10,10*4=40, 40+ 5=45, 6*5=30 (4 5 6 45 30)
**4/ 7+5=12, 12*5=60, 60+6=66, 6*7= 42 (5 6 7 66 42 )
**5/ 8+6=14, 14*6=84, 84+7=91, 8*7=56 (6 7 8 91 56 )
 
 
 

91

91

15
28
45
66
?
28-15 = 13
45-28 = 17
66-45 = 21
now take the answers
17-13=4
21-17=4
now take
21 + 4 + 66=91
91 is the answer

Your series seems to consist of 5 interleaved series. The first three are linear and the last two are quadratic. The nth term of each series is computed as
  {1+n, 2+n, 3+n, 6+7n+2n^2, 6+5n+n^2}
The first 7 terms in each series, interleaved as you have them, are
  2, 3, 4, 15, 12, 3, 4, 5, 28, 20, 4, 5, 6, 45, 30, 5, 6, 7, 66, 42, 6, 7, 8, 91, 56, 7, 8, 9, 120, 72, 8, 9, 10, 153, 90

Your missing number is 91.
It is computed using the above formula for n=5.

---

For oddball series like this, I find it convenient to start by plotting the points and looking for patterns. There are two distinct quadratic patterns. The terms of them are 5 numbers apart in the given list, suggesting 5 interleaved series. Writing every 5th number in its own series, we get
  {2, 3, 4, 5, 6}, {3, 4, 5, 6, 7}, {4, 5, 6, 7, 8}, {15, 28, 45, 66, x}, {12, 20, 30, 42, 56}
Here's the plot.

11

21

2 - 3 - 4 - 15 - 12 3 - 4 - 5 - 28 - 20 4 - 5 - 6 - 45 - 30 5 - 6 - 7 - 66 - 42 6 - 7 - 8 - 83 - 56