Answer:
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
