a sequence tn, is defined as all natural numbera that are neither perfect squares nor perfect cubes. what are the first ten terms?
tn = {2, 3, 5, 6, 7, 10, 11, 12, 13, 14}
is this correct?
Yes, that looks correct.
To find the first ten terms of the sequence tn, which consists of natural numbers that are neither perfect squares nor perfect cubes, one approach is to go through the natural numbers starting from 1 and check each number to see if it is a perfect square or a perfect cube.
To determine if a number is a perfect square, we can take the square root of the number and check if the result is an integer. If it is, then the number is a perfect square. For example, √4 = 2, so 4 is a perfect square.
Similarly, to determine if a number is a perfect cube, we can take the cube root of the number and check if the result is an integer. For example, ∛8 = 2, so 8 is a perfect cube.
Starting with the first natural number 1, we can go through each number, checking if it is neither a perfect square nor a perfect cube, until we have found the first ten terms:
1 is a perfect square (1^2) and a perfect cube (1^3), so it is not part of the sequence.
2 is neither a perfect square nor a perfect cube, so it is part of the sequence.
3 is neither a perfect square nor a perfect cube, so it is part of the sequence.
4 is a perfect square (2^2) and a perfect cube (2^3), so it is not part of the sequence.
5 is neither a perfect square nor a perfect cube, so it is part of the sequence.
6 is neither a perfect square nor a perfect cube, so it is part of the sequence.
7 is neither a perfect square nor a perfect cube, so it is part of the sequence.
8 is a perfect square (2^2) and a perfect cube (2^3), so it is not part of the sequence.
9 is a perfect square (3^2) and a perfect cube (3^3), so it is not part of the sequence.
10 is neither a perfect square nor a perfect cube, so it is part of the sequence.
Therefore, the first ten terms of the sequence tn are: {2, 3, 5, 6, 7, 10, 11, 12, 13, 14}.