The sequence 2, 3, 5, 6, 7, 10, 11, contains all the positive integers from least to greatest that are neither squares nor cubes nor perfect fifth powers (in the form of x⁵, where x is an integer). What is the term of the sequence?

1041 is the 1000th term

Step-by-step explanation:

First of all, let B(n) be the number of integers in range {1,2,…,n} that are not squares, cubes or fifth powers.

Now, we need to find the first number such that B(n)=1000.

A formula for B(n) can be obtained with inclusion exclusion principle.

Thus,

B(n)= n − (√n) − (∛n) − (5√n) + (6√n) +(10√n) + (15√n) − (30√n)

This is very close to n.

Let's try this method;

Take N(o) = 1000 and if we take

N(i + 1) = Ni + (1000−B(Ni)),we'll notice it guarantees B(N(i−1)) < 1000

Thus, we can use the method;

At N(o) = 1000 ;

B(N(o)) = 1000 − 31 − 10 − 3 + 3 + 1 + 1 − 1 = 960

At N1 = 1040 ; B(N1)=1040−32−10−4+3+2+1−1 = 999

At N2 = 1041 ; B(N1)=1041−32−10−4+3+2+1−1=1000

Thus, 1041 is the 1000th term