Determine the number that correctly completes the following series.

3, 11, 33, 75, 143, ?

The correct answer is 243.

To determine the number that completes the series 3, 11, 33, 75, 143, ?, we can analyze the pattern formed by the differences between consecutive terms.

Step 1: Calculate the first-level differences between consecutive terms.

$11-3=8$

$33-11=22$

$75-33=42$

$143-75=68$

The sequence of first-level differences is: 8, 22, 42, 68.

Step 2: Calculate the second-level differences (the difference between successive first-level differences).

$22-8=14$

$42-22=20$

$68-42=26$

The second-level differences are: 14, 20, 26. Notice that these values increase consistently by 6 each time:

$20-14=6$

$26-20=6$

Step 3: Calculate the next second-level difference.
Add 6 to the last second-level difference (26):

$26+6=32$

Step 4: Calculate the next first-level difference.
Add 32 to the last first-level difference (68):

$68+32=100$

Step 5: Find the missing term in the series.
Finally, add 100 to the last term of the given series (143):

$143+100=243$

Thus, the number that correctly completes the series is 243.