To find the next number in the sequence 3, 6, 11, 18, 27, we need to identify the pattern or rule governing the progression of numbers.
Let's examine the differences between consecutive terms:
- The difference between the first term (3) and the second term (6) is: \(6 - 3 = 3\)
- The difference between the second term (6) and the third term (11) is: \(11 - 6 = 5\)
- The difference between the third term (11) and the fourth term (18) is: \(18 - 11 = 7\)
- The difference between the fourth term (18) and the fifth term (27) is: \(27 - 18 = 9\)
From the calculated differences, we get the sequence of differences: 3, 5, 7, 9. Here, we observe that the differences between consecutive numbers are increasing by 2 each time.
The next difference should be: \(9 + 2 = 11\)
Adding this next difference to the last number in the sequence gives us the next term:
\(27 + 11 = 38\)
Therefore, the next number in the sequence is 38.
Let's justify why the given options are correct or incorrect:
- 32: The difference growth pattern does not match the sequence changes.
- 36: By following the pattern of differences, we see 36 is skipped by two more than necessary difference.
- 38: This correctly fits our calculation based on the sequence pattern.
- 40: Exceeds the required difference pattern. It would follow a different difference growth.
Hence, the correct answer is
38