Bogme « Question 44.

The sequence is formed by adding the cubes of consecutive integers starting from 1 to the previous term, but with a specific offset or pattern in the differences. Specifically, the differences between terms are 3, 16, 33, 52, 75, 104, 135, which are derived from a second-order difference sequence that is linear.

Step-by-step Derivation:
Step 1: Calculate the first differences between consecutive terms:
16 - 13 = 3
32 - 16 = 16
65 - 32 = 33
117 - 65 = 52
192 - 117 = 75
296 - 192 = 104
431 - 296 = 135

Step 2: Calculate the second differences (differences of the differences):
16 - 3 = 13
33 - 16 = 17
52 - 33 = 19
75 - 52 = 23
104 - 75 = 29
135 - 104 = 31

Step 3: Analyze the second differences: 13, 17, 19, 23, 29, 31. These are consecutive prime numbers starting from 13.

Step 4: Find the next prime number after 31, which is 37.

Step 5: Calculate the next first difference by adding the next prime to the last first difference:
135 + 37 = 172

Step 6: Calculate the next term in the original sequence by adding this difference to the last term:
431 + 172 = 603

Wait, let me re-verify the prime sequence: 13, 17, 19, 23, 29, 31. The next prime is indeed 37. 431 + 172 = 603.
Looking at the options: A) oo, B) 602, C) 604, D) 603 i.
Option D is '603 i', which likely contains a typo for '603'. However, checking the second differences again: 13, 17, 19, 23, 29, 31. If the pattern was simply odd numbers or a different progression, let's check: 13, 17, 19, 23, 29, 31... these are primes.
If we check the options again, 603 is listed as D. If the question intended 603, D is the answer. However, if we check for a different pattern: 13, 17, 19, 23, 29, 31... if the next difference was 36 (not prime), it would be 602. Let's re-examine the second differences: 13, 17, 19, 23, 29, 31. These are exactly the primes from 13 to 31. The next prime is 37. 431 + 135 + 37 = 603.
Given the options, 603 is the mathematical result. Option D is '603 i'. Option B is '602'. Let's re-calculate: 431 + 171 = 602. 171 - 135 = 36. 36 is not prime.
Actually, looking at the sequence again: 13, 17, 19, 23, 29, 31... the gaps are 4, 2, 4, 6, 2. This is the gap between primes. The next gap is 6 (31 to 37).
If the answer is 602, the difference is 171, and the second difference is 36. If the answer is 603, the difference is 172, and the second difference is 37.
Since 37 is the next prime, 603 is the logical answer. Option D is '603 i'. In many test banks, 'i' is a typo. However, if the sequence was 13, 17, 19, 23, 27, 31 (arithmetic-like), the next would be 35. 135+35=170. 431+170=601.
Re-evaluating the second differences: 13, 17, 19, 23, 29, 31. These are primes. Next prime is 37. 431 + 135 + 37 = 603. Option D is 603.