What will come in place of * in the sequence 3, 14, 39, 84, *, 258?

The correct option is 155.

To find the missing number in the sequence 3, 14, 39, 84, *, 258, let us analyze the differences between consecutive terms:
First difference between 14 and 3: 14 - 3 = 11
Second difference between 39 and 14: 39 - 14 = 25
Third difference between 84 and 39: 84 - 39 = 45

Now, let us examine the sequence of these differences: 11, 25, 45. We look at the differences of these differences (the second-order differences):
Difference between 25 and 11: 25 - 11 = 14
Difference between 45 and 25: 45 - 25 = 20

We can see that the second-order differences (14, 20) increase by 6. If this pattern continues, the next second-order difference should be:
20 + 6 = 26

Using this next second-order difference of 26, the next difference between the terms of the main sequence will be:
45 + 26 = 71

Therefore, the missing term (*) in the sequence can be calculated by adding this difference to the previous term (84):
84 + 71 = 155

To verify this pattern, let us calculate the next term in the sequence using the same logic:
The next second-order difference would be: 26 + 6 = 32
The next difference between the terms of the main sequence would be: 71 + 32 = 103
The term after the missing term should be: 155 + 103 = 258

Since this matches the final term (258) in the given sequence, the logic is consistent, and the correct number in place of * is indeed 155.