What number would replace question mark (?) in the series given below?

3, 8, 17, 32, 57, ?, 177

The correct option is: 100.

To find the missing number in the given series, let us analyze the differences between consecutive terms:

First level of differences:
$8-3=5$
$17-8=9$
$32-17=15$
$57-32=25$

Now, let us find the differences between these first-level differences (the second level of differences):

Second level of differences:
$9-5=4$
$15-9=6$
$25-15=10$

If we look closely at the sequence of second differences (4, 6, 10, ...), we can observe a pattern in the differences between them (the third level of differences):

Third level of differences:
$6-4=2$
$10-6=4$

The differences in the third level are doubling (2, 4, ...). Following this pattern, the next third-level differences should be 8 and then 16.

Using this logic, we can work backwards to find the missing terms:

The next second-level difference: $10+8=18$
The subsequent second-level difference: $18+16=34$

Now, we find the next first-level differences by adding these values to the previous first-level differences:

The next first-level difference: $25+18=43$
The subsequent first-level difference: $43+34=77$

Finally, we calculate the missing term (?) in our original sequence:

The missing number replacing the question mark (?): $57+43=100$

We can verify this by checking if it leads to the final number in the series (177) using the subsequent difference of 77:

Verification: $100+77=177$

Since the calculations match the given sequence, the number that replaces the question mark is indeed 100.