1512, 1524, 1540, 1563, ?, 1642
What will come in the place of question (?) mark in following number series:

The correct option is 1596.

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

1st difference: 1524 - 1512 = 12

2nd difference: 1540 - 1524 = 16

3rd difference: 1563 - 1540 = 23

Now, let us examine how these differences (12, 16, 23) relate to the numbers in the series. Notice that each difference is formed by adding the digits of the preceding number to that number itself:

For the first term, 1512:

Sum of digits = 1 + 5 + 1 + 2 = 9

Wait, let's look at another pattern in the differences:

Differences: 12, 16, 23

Difference of differences:

16 - 12 = 4

23 - 16 = 7

Alternatively, let's look at the digits of each term:

1512 → 1512 + (1 + 5 + 1 + 2 + 3) = 1524?

Let's check the sum of digits of 1512:

1 + 5 + 1 + 2 = 9; 1512 + 9 + 3 = 1524 (difference 12)

1524: sum of digits = 1 + 5 + 2 + 4 = 12; 1524 + 12 + 4 = 1540 (difference 16)

1540: sum of digits = 1 + 5 + 4 + 0 = 10; 1540 + 10 + 13 = 1563 (difference 23)

Let's check another pattern for differences: 12, 16, 23, ?, ?

Second level difference: 4, 7, 11, 16... (adding prime or Fibonacci-like sequence or +3 pattern?)

If 2nd level differences are: 4, 7, 10, 13...

12 + 4 = 16

16 + 7 = 23

23 + 10 = 33

Then next term = 1563 + 33 = 1596.

Let's verify the final term using this pattern:

33 + 13 = 46

1596 + 46 = 1642

This perfectly matches the last term in the series (1642)!

Step-by-step breakdown:

1. Find the difference between consecutive terms:

1524 - 1512 = 12

1540 - 1524 = 16

1563 - 1540 = 23

2. Find the second-level differences (differences between differences):

16 - 12 = 4

23 - 16 = 7

3. Notice the pattern in the second-level differences (+3 in each step):

4, 7, 10, 13

4. Calculate the next first-level difference:

23 + 10 = 33

5. Add 33 to the fourth term (1563) to find the missing term:

$1563+33=1596$

6. Verification of the final term:

Next difference = 33 + 13 = 46

$1596+46=1642$

Therefore, the missing number in the place of question mark (?) is 1596.