Directions: Determine the missing number that should replace the question mark (?) in the series provided below.

?, 8, 19, 61, 249, 1251

The correct answer is 6.

To determine the missing number in the given series, let us analyze the logical relationship between consecutive terms from left to right:

?, 8, 19, 61, 249, 1251

Let us examine the operations that connect the known terms:

From the 3rd term (19) to the 4th term (61):
$19\times 3+4=57+4=61$

From the 4th term (61) to the 5th term (249):
$61\times 4+5=244+5=249$

From the 5th term (249) to the 6th term (1251):
$249\times 5+6=1245+6=1251$

From the above calculations, we can establish the general pattern rule for this series:

$\text{Next term}=(\text{Current term}\times n)+(n+1)$

where $n$ starts at 1 for the first transition and increases by 1 for each subsequent step.

Now, let us use this rule to find the missing first term represented by $?$:

For the 1st transition (from $?$ to 8), $n=1$:
$?\times 1+2=8$
$?+2=8$
$?=8-2$
$?=6$

To verify, let us test the 2nd transition (from 8 to 19) where $n=2$:
$8\times 2+3=16+3=19$

The pattern holds true across the entire series. Therefore, the missing number that should replace the question mark is 6.