Fun and Hard:

I don't know any specific rule for solving this problem.just to share how I did.

There are 27 numbers which are either square or cube of a positive integer below 500.

so 500-27=473 means you have got 473 possibilities.now

start counting from 501 for the rest 27 numbers while you have to ignore a number (512=8^3) in your course.you will reach the 500th term at 528.

let me clarify this a bit. suppose you need the 20th term of the sequence

1,2,3,4,5...................,18,19,20,21,22,23,24,25,26

there are 5 numbers below 20 which are either square or cube of a positive integer.

20-5=15.you got(2,3,5,6,7,10,11,12,13,14,15,17,18,19,20) but you still need 5 more numbers. so count from 21,22,23,24,26. here 26 is the 20th term

Considering numbers less than or equal to 500, we need to find the number of squares and cubes.

You can find the number of squares or cubes by either getting the square root and cube root of 500 and rounding down to the nearest integer, or by calculating the nearest squares.

Number of squares = 22 Number of cubes = 7

We need to find the number of Squares or Cubes. We can only add the number of squares and the number of cubes if they are mutually exclusive (no common elements), and unfortunately we have numbers which are both squares and cubes [1 = 1^2, 1^3 and 64 = 8^2, 4^3]

To determine the number of cubes or squares we simply have to use this equation:

N(square or cube) = N(square) + N(cube) - N(square and cube)

Subtracting the count of numbers that are both square and cube is necessary to eliminate double counting.

N(square or cube) = 22 + 7 - 2 = 27

500+27 = 527

After doing so, we need to check for squares and cubes in the range of 501 and 527 (inclusive). We could then find that 512 = 8^3 so we need to add another number and it becomes 528 and because 528 is neither a square nor a cube it is the final answer.

Sidenote: If it were the 501st term, it would become 530 because 529 = 23^2

This was my approach: Because the smallest square over 500 is 23^2 (529) and 500 numbers in the sequence are less than 529; 528 would be the answer as it is just less than 529 and is not a cube or a square.