Question

cai writes down the list of positive integers, excluding squares and cubes. his sequence starts \[2, \ 3, \ 5, \ 6, \ 7, \ 10, \ 11, \ \dots.\]what is the $1000$th term in cai's list?

Answer 1

The 1000th term in Cai's list is the number that comes after the 1000th positive integer and the first 998 perfect square and perfect cube numbers.

Step-by-step explanation:

The given sequence starts with the positive integers excluding squares and cubes: 2, 3, 5, 6, 7, 10, 11, ....

To find the 1000th term in Cai's list, we need to identify the pattern of the sequence. The sequence skips the perfect squares (1, 4, 9, 16, ...) and the perfect cubes (1, 8, 27, 64, ...).

Since Cai's sequence is skipping numbers, we can observe that every perfect square and perfect cube number is skipped. Therefore, the 1000th term in Cai's list is the number that comes after the 1000th positive integer and the first 998 perfect square and perfect cube numbers.