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 number that replaces the question mark (?), let us analyze the differences between consecutive terms in the given series:

The given series is: 3, 8, 17, 32, 57, ?, 177

Let's find the difference between consecutive terms:
First difference:
$8-3=5$
Second difference:
$17-8=9$
Third difference:
$32-17=15$
Fourth difference:
$57-32=25$

Now, let us examine the sequence of these differences: 5, 9, 15, 25.
We can find the differences between these differences (the second-level differences):
Difference between 9 and 5:
$9-5=4$
Difference between 15 and 9:
$15-9=6$
Difference between 25 and 15:
$25-15=10$

The second-level differences are: 4, 6, 10.
Looking at this second-level difference pattern, we can see a relationship of multiplying by 2 and then subtracting a value, or adding consecutive even numbers with an increasing gap:
Let's find the differences of the second-level differences (third-level differences):
$6-4=2$
$10-6=4$
The differences of the second-level differences are 2, 4, which suggests the next third-level difference should be 8 (doubling the previous difference) or 6 (increasing by 2).
Let's test the doubling pattern for third-level differences: 2, 4, 8, 16...
If the next third-level difference is 8:
The next second-level difference would be:
$10+8=18$
The next first-level difference would be:
$25+18=43$
The missing term (?) would then be:
$57+43=100$

Let us verify if this rule holds for the final term (177):
The next third-level difference would be:
$8\times 2=16$
The next second-level difference would be:
$18+16=34$
The next first-level difference would be:
$43+34=77$
The next term after 100 would be:
$100+77=177$
Since this matches the last term in the given series, the pattern is verified and correct.

Therefore, the number that replaces the question mark is 100.